## Chapter Review Exercises

### Solve Quadratic Equations Using the Square Root Property

Exercise (PageIndex{1}) Solve Quadratic Equations of the Form (ax^{2}=k) Using the Square Root Property

In the following exercises, solve using the Square Root Property.

- (y^{2}=144)
- (n^{2}-80=0)
- (4 a^{2}=100)
- (2 b^{2}=72)
- (r^{2}+32=0)
- (t^{2}+18=0)
- (frac{2}{3} w^{2}-20=30)
- (5 c^{2}+3=19)

**Answer**1. (y=pm 12)

3. (a=pm 5)

5. (r=pm 4 sqrt{2} i)

7. (w=pm 5 sqrt{3})

Exercise (PageIndex{2}) Solve Quadratic Equations of the Form (a(x-h)^{2}=k) Using the Square Root Property

In the following exercises, solve using the Square Root Property.

- ((p-5)^{2}+3=19)
- ((u+1)^{2}=45)
- (left(x-frac{1}{4} ight)^{2}=frac{3}{16})
- (left(y-frac{2}{3} ight)^{2}=frac{2}{9})
- ((n-4)^{2}-50=150)
- ((4 c-1)^{2}=-18)
- (n^{2}+10 n+25=12)
- (64 a^{2}+48 a+9=81)

**Answer**1. (p=-1,9)

3. (x=frac{1}{4} pm frac{sqrt{3}}{4})

5. (n=4 pm 10 sqrt{2})

7. (n=-5 pm 2 sqrt{3})

### Solve Quadratic Equations by Completing the Square

Exercise (PageIndex{3}) Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

- (x^{2}+22 x)
- (m^{2}-8 m)
- (a^{2}-3 a)
- (b^{2}+13 b)

**Answer**1. ((x+11)^{2})

3. (left(a-frac{3}{2} ight)^{2})

Exercise (PageIndex{4}) Solve Quadratic Equations Using Completing the Square

In the following exercises, solve by completing the square.

- (d^{2}+14 d=-13)
- (y^{2}-6 y=36)
- (m^{2}+6 m=-109)
- (t^{2}-12 t=-40)
- (v^{2}-14 v=-31)
- (w^{2}-20 w=100)
- (m^{2}+10 m-4=-13)
- (n^{2}-6 n+11=34)
- (a^{2}=3 a+8)
- (b^{2}=11 b-5)
- ((u+8)(u+4)=14)
- ((z-10)(z+2)=28)

**Answer**1. (d=-13,-1)

3. (m=-3 pm 10 i)

5. (v=7 pm 3 sqrt{2})

7. (m=-9,-1)

9. (a=frac{3}{2} pm frac{sqrt{41}}{2})

11. (u=-6 pm 2 sqrt{2})

### Solve Quadratic Equations of the Form (ax^{2}+bx+c=0) by Completing the Square

Exercise (PageIndex{5}) Solve Quadratic Equations of the Form (ax^{2}+bx+c=0) by Completing the Square

In the following exercises, solve by completing the square.

- (3 p^{2}-18 p+15=15)
- (5 q^{2}+70 q+20=0)
- (4 y^{2}-6 y=4)
- (2 x^{2}+2 x=4)
- (3 c^{2}+2 c=9)
- (4 d^{2}-2 d=8)
- (2 x^{2}+6 x=-5)
- (2 x^{2}+4 x=-5)

**Answer**1. (p=0,6)

3. (y=-frac{1}{2}, 2)

5. (c=-frac{1}{3} pm frac{2 sqrt{7}}{3})

7. (x=frac{3}{2} pm frac{1}{2} i)

Exercise (PageIndex{6}) Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

- (4 x^{2}-5 x+1=0)
- (7 y^{2}+4 y-3=0)
- (r^{2}-r-42=0)
- (t^{2}+13 t+22=0)
- (4 v^{2}+v-5=0)
- (2 w^{2}+9 w+2=0)
- (3 m^{2}+8 m+2=0)
- (5 n^{2}+2 n-1=0)
- (6 a^{2}-5 a+2=0)
- (4 b^{2}-b+8=0)
- (u(u-10)+3=0)
- (5 z(z-2)=3)
- (frac{1}{8} p^{2}-frac{1}{5} p=-frac{1}{20})
- (frac{2}{5} q^{2}+frac{3}{10} q=frac{1}{10})
- (4 c^{2}+4 c+1=0)
- (9 d^{2}-12 d=-4)

**Answer**1. (x=frac{1}{4}, 1)

3. (r=-6,7)

5. (v=frac{-1 pm sqrt{21}}{8})

7. (m=frac{-4 pm sqrt{10}}{3})

9. (a=frac{5}{12} pm frac{sqrt{23}}{12} i)

11. (u=5 pm sqrt{21})

13. (p=frac{4 pm sqrt{5}}{5})

15. (c=-frac{1}{2})

Exercise (PageIndex{7}) Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions for each quadratic equation.

- (9 x^{2}-6 x+1=0)
- (3 y^{2}-8 y+1=0)
- (7 m^{2}+12 m+4=0)
- (5 n^{2}-n+1=0)

- (5 x^{2}-7 x-8=0)
- (7 x^{2}-10 x+5=0)
- (25 x^{2}-90 x+81=0)
- (15 x^{2}-8 x+4=0)

**Answer**1.

- (1)
- (2)
- (2)
- (2)

Exercise (PageIndex{8}) Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

- (16 r^{2}-8 r+1=0)
- (5 t^{2}-8 t+3=9)
- (3(c+2)^{2}=15)

- (4 d^{2}+10 d-5=21)
- (25 x^{2}-60 x+36=0)
- (6(5 v-7)^{2}=150)

**Answer**1.

- Factor
- Quadratic Formula
- Square Root

### Solve Equations in Quadratic Form

Exercise (PageIndex{9}) Solve Equations in Quadratic Form

In the following exercises, solve.

- (x^{4}-14 x^{2}+24=0)
- (x^{4}+4 x^{2}-32=0)
- (4 x^{4}-5 x^{2}+1=0)
- ((2 y+3)^{2}+3(2 y+3)-28=0)
- (x+3 sqrt{x}-28=0)
- (6 x+5 sqrt{x}-6=0)
- (x^{frac{2}{3}}-10 x^{frac{1}{3}}+24=0)
- (x+7 x^{frac{1}{2}}+6=0)
- (8 x^{-2}-2 x^{-1}-3=0)

**Answer**1. (x=pm sqrt{2}, x=pm 2 sqrt{3})

3. (x=pm 1, x=pm frac{1}{2})

5. (x=16)

7. (x=64, x=216)

9. (x=-2, x=frac{4}{3})

### Solve Applications of Quadratic Equations

Exercise (PageIndex{10}) Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

- Find two consecutive odd numbers whose product is (323).
- Find two consecutive even numbers whose product is (624).
- A triangular banner has an area of (351) square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.
- Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is (70) square inches. Find the height and width of the case.
- A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is (5) feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.

*Figure 9.E.1*

6. A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

7. The front walk from the street to Pam’s house has an area of (250) square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.

8. For Sophia’s graduation party, several tables of the same width will be arranged end to end to give serving table with a total area of (75) square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth . Round answer to the nearest tenth.

9. A ball is thrown vertically in the air with a velocity of (160) ft/sec. Use the formula (h=-16 t^{2}+v_{0} t) to determine when the ball will be (384) feet from the ground. Round to the nearest tenth.

10. The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of (5) hours and each way the trip was (360) miles. If the plane was flying at (150) mph, what was the speed of the wind that affected the plane?

11. Ezra kayaked up the river and then back in a total time of (6) hours. The trip was (4) miles each way and the current was difficult. If Roy kayaked at a speed of (5) mph, what was the speed of the current?

12. Two handymen can do a home repair in (2) hours if they work together. One of the men takes (3) hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?

**Answer**2. Two consecutive even numbers whose product is (624) are (24) and (26), and (−24) and (−26).

4. The height is (14) inches and the width is (10) inches.

6. The length of the diagonal is (3.6) feet.

8. The width of the serving table is (4.7) feet and the length is (16.1) feet.

10. The speed of the wind was (30) mph.

12. One man takes (3) hours and the other man (6) hours to finish the repair alone.

### Graph Quadratic Functions Using Properties

Exercise (PageIndex{11}) Recognize the Graph of a Quadratic Function

In the following exercises, graph by plotting point.

- Graph (y=x^{2}-2)
- Graph (y=-x^{2}+3)

**Answer**2.

Exercise (PageIndex{12}) Recognize the Graph of a Quadratic Function

In the following exercises, determine if the following parabolas open up or down.

- (y=-3 x^{2}+3 x-1)
- (y=5 x^{2}+6 x+3)

- (y=x^{2}+8 x-1)
- (y=-4 x^{2}-7 x+1)

**Answer**2.

- Up
- Down

Exercise (PageIndex{13}) Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find

- The equation of the axis of symmetry
- The vertex
- (y=-x^{2}+6 x+8)
- (y=2 x^{2}-8 x+1)

**Answer**2. (x=2) ; ((2,-7))

Exercise (PageIndex{14}) Find the Intercepts of a Parabola

In the following exercises, find the (x)- and (y)-intercepts.

- (y=x^{2}-4x+5)
- (y=x^{2}-8x+15)
- (y=x^{2}-4x+10)
- (y=-5x^{2}-30x-46)
- (y=16x^{2}-8x+1)
- (y=x^{2}+16x+64)

**Answer**2. (egin{array}{l}{y :(0,15)} {x :(3,0),(5,0)}end{array})

4. (egin{array}{l}{y :(0,-46)} {x : ext { none }}end{array})

6. (egin{array}{l}{y :(0,-64)} {x :(-8,0)}end{array})

#### Graph Quadratic Functions Using Properties

Exercise (PageIndex{15}) Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

- (y=x^{2}+8 x+15)
- (y=x^{2}-2 x-3)
- (y=-x^{2}+8 x-16)
- (y=4 x^{2}-4 x+1)
- (y=x^{2}+6 x+13)
- (y=-2 x^{2}-8 x-12)

**Answer**2.

4.

6.

Exercise (PageIndex{16}) Solve Maximum and Minimum Applications

In the following exercises, find the minimum or maximum value.

- (y=7 x^{2}+14 x+6)
- (y=-3 x^{2}+12 x-10)

**Answer**2. The maximum value is (2) when (x=2).

Exercise (PageIndex{17}) Solve Maximum and Minimum Applications

In the following exercises, solve. Rounding answers to the nearest tenth.

- A ball is thrown upward from the ground with an initial velocity of (112) ft/sec. Use the quadratic equation (h=-16 t^{2}+112 t) to find how long it will take the ball to reach maximum height, and then find the maximum height.
- A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using (180) feet of fencing on three sides of the yard. The quadratic equation (A=-2 x^{2}+180 x) gives the area, (A), of the yard for the length, (x), of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

**Answer**2. The length adjacent to the building is (90) feet giving a maximum area of (4,050) square feet.

### Graph Quadratic Functions Using Transformations

Exercise (PageIndex{18}) Graph Quadratic Functions of the Form (f(x)=x^{2}+k)

In the following exercises, graph each function using a vertical shift.

- (g(x)=x^{2}+4)
- (h(x)=x^{2}-3)

**Answer**2.

Exercise (PageIndex{19}) Graph Quadratic Functions of the Form (f(x)=x^{2}+k)

In the following exercises, graph each function using a horizontal shift.

- (f(x)=(x+1)^{2})
- (g(x)=(x-3)^{2})

**Answer**2.

Exercise (PageIndex{20}) Graph Quadratic Functions of the Form (f(x)=x^{2}+k)

In the following exercises, graph each function using transformations.

- (f(x)=(x+2)^{2}+3)
- (f(x)=(x+3)^{2}-2)
- (f(x)=(x-1)^{2}+4)
- (f(x)=(x-4)^{2}-3)

**Answer**2.

4.

Exercise (PageIndex{21}) Graph Quadratic Functions of the Form (f(x)=ax^{2})

In the following exercises, graph each function.

- (f(x)=2x^{2})
- (f(x)=-x^{2})
- (f(x)=frac{1}{2} x^{2})

**Answer**2.

Exercise (PageIndex{22}) Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the (f(x)=a(x-h)^{2}+k) form by completing the square.

- (f(x)=2 x^{2}-4 x-4)
- (f(x)=3 x^{2}+12 x+8)

**Answer**1. (f(x)=2(x-1)^{2}-6)

Exercise (PageIndex{23}) Graph Quadratic Functions Using Transformations

In the following exercises,

- Rewrite each function in (f(x)=a(x−h)^{2}+k) form
- Graph it by using transformations
- (f(x)=3 x^{2}-6 x-1)
- (f(x)=-2 x^{2}-12 x-5)
- (f(x)=2 x^{2}+4 x+6)
- (f(x)=3 x^{2}-12 x+7)

**Answer**1.

- (f(x)=3(x-1)^{2}-4)
*Figure 9.E.13*

3.

- (f(x)=2(x+1)^{2}+4)
*Figure 9.E.14*

Exercise (PageIndex{24}) Graph Quadratic Functions Using Transformations

In the following exercises,

- Rewrite each function in (f(x)=a(x−h)^{2}+k) form
- Graph it using properties
- (f(x)=-3 x^{2}-12 x-5)
- (f(x)=2 x^{2}-12 x+7)

**Answer**1.

- (f(x)=-3(x+2)^{2}+7)
*Figure 9.E.15*

Exercise (PageIndex{25}) Find a Quadratic Function From its Graph

In the following exercises, write the quadratic function in (f(x)=a(x−h)^{2}+k) form.

*Figure 9.E.16**Figure 9.E.17*

**Answer**1. (f(x)=(x+1)^{2}-5)

### Solve Quadratic Inequalities

Exercise (PageIndex{26}) Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

- (x^{2}-x-6>0)
- (x^{2}+4 x+3 leq 0)
- (-x^{2}-x+2 geq 0)
- (-x^{2}+2 x+3<0)

**Answer**1.

*Figure 9.E.18*- ((-infty,-2) cup(3, infty))

3.

*Figure 9.E.19*- ([-2,1])

Exercise (PageIndex{27}) Solve Quadratic Inequalities Graphically

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

- (x^{2}-6 x+8<0)
- (x^{2}+x>12)
- (x^{2}-6 x+4 leq 0)
- (2 x^{2}+7 x-4>0)
- (-x^{2}+x-6>0)
- (x^{2}-2 x+4 geq 0)

**Answer**1. ((2,4))

3. ([3-sqrt{5}, 3+sqrt{5}])

5. no solution

## Practice Test

Exercise (PageIndex{28})

- Use the Square Root Property to solve the quadratic equation (3(w+5)^{2}=27).
- Use Completing the Square to solve the quadratic equation (a^{2}-8 a+7=23).
- Use the Quadratic Formula to solve the quadratic equation (2 m^{2}-5 m+3=0).

**Answer**1. (w=-2, w=-8)

3. (m=1, m=frac{3}{2})

Exercise (PageIndex{29})

Solve the following quadratic equations. Use any method.

- (2 x(3 x-2)-1=0)
- (frac{9}{4} y^{2}-3 y+1=0)

**Answer**2. (y=frac{2}{3})

Exercise (PageIndex{30})

Use the discriminant to determine the number and type of solutions of each quadratic equation.

- (6 p^{2}-13 p+7=0)
- (3 q^{2}-10 q+12=0)

**Answer**2. (2) complex

Exercise (PageIndex{31})

Solve each equation.

- (4 x^{4}-17 x^{2}+4=0)
- (y^{frac{2}{3}}+2 y^{frac{1}{3}}-3=0)

**Answer**2. (y=1, y=-27)

Exercise (PageIndex{32})

For each parabola, find

- Which direction it opens
- The equation of the axis of symmetry
- The vertex
- The (x)
*-*and (y)-intercepts - The maximum or minimum value
- (y=3 x^{2}+6 x+8)
- (y=-x^{2}-8 x+16)

**Answer**2.

- down
- (x=-4)
- ((-4,0))
- (y: (0,16); x: (-4,0))
- minimum value of (-4) when (x=0)

Exercise (PageIndex{33})

Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.

- (f(x)=x^{2}+6 x+9)
- (f(x)=-2 x^{2}+8 x+4)

**Answer**2.

Exercise (PageIndex{34})

In the following exercises, graph each function using transformations.

- (f(x)=(x+3)^{2}+2)
- (f(x)=x^{2}-4 x-1)

**Answer**2.

*Figure 9.E.21*

Exercise (PageIndex{35})

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

- (x^{2}-6 x-8 leq 0)
- (2 x^{2}+x-10>0)

**Answer**2. (left(-infty,-frac{5}{2} ight) cup(2, infty))

Exercise (PageIndex{36})

Model the situation with a quadratic equation and solve by any method.

- Find two consecutive even numbers whose product is (360).
- The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)

**Answer**2. A water balloon is launched upward at the rate of (86) ft/sec. Using the formula (h=-16 t^{2}+86 t) find how long it will take the balloon to reach the maximum height, and then find the maximum height. Round to the nearest tenth.

Some of the words in parentheses require the addition of the letter *e* others are correct as they stand.

- Gus is (tru-ly) sorry for keeping you awake last night.
- We were criticized (sever-ly) by the sewing circle.
- The shed was (complet-ly) demolished.
- Merdine was (sincer-ly) grateful for the reprieve.
- The Simpsons are (argu-ing) again.
- They (argu-d) last night for hours.
- When is Mr. Wolfe (com-ing) home.
- Maya is (writ-ing) her autobiography.
- Mr. White is (judg-ing) the essay contest.
- Be (car-ful) when you light the furnace.

## 16.1: Acids and Bases: A Brief Review

### Conceptual Problems

- Identify the conjugate acid&ndashbase pairs in each equilibrium.
- (HSO^&minus_<4>,(aq)+H_2O,(l) ightleftharpoons SO^<2&minus>_<4>,(aq)+H_3O^<+>,(aq))
- (C_<3>H_<7>NO_<2>,(aq)+H_<3>O^<+>,(aq) ightleftharpoons C_<3>H_<8>NO^<+>_<2>,(aq)+H_<2>O,(l))
- (CH_<3>O_<2>H,(aq)+NH_<3>,(aq) ightleftharpoons CH_<3>CO^<&minus>_<2>,(aq)+NH^<+>_<4>,(aq))
- (SbF_<5>,(aq)+2,HF,(aq) ightleftharpoons H_<2>F^<+>,(aq)+SbF^<&minus>_<6>,(aq))

- (HF,(aq)+H_<2>O,(l) ightleftharpoons H_3O^<+>,(aq)+F^<&minus>,(aq))
- (CH_3CH_2NH_<2>,(aq)+H_<2>O,(l) ightleftharpoons CH_3CH_2NH^<+>_<3>,(aq)+OH^<&minus>,(aq))
- (C_3H_7NO_<2>,(aq)+OH^<&minus>,(aq) ightleftharpoons C_3H_6NO^<&minus>_<2>,(aq)+H_<2>O,(l))
- (CH_3CO_2H,(aq)+2,HF,(aq) ightleftharpoons CH_3C(OH)_<2>^<+>,(aq)+HF^<&minus>_<2>,(aq))

- (HCO^&minus_<3>,(aq)+H_2O,(l) ightleftharpoons CO^<2&minus>_<3>,(aq)+H_3O^<+>,(aq))
- (formicacid,(aq)+H_2O,(l) ightleftharpoons formate,(aq)+H_3O^+,(aq))
- (H_3PO_<4>,(aq)+H_2O,(l) ightleftharpoons H_2PO^&minus_<4>,(aq)+H_3O^+,(aq))

- (OCH^<&minus>_<3>,(aq)+H_2O,(l) ightleftharpoons HOCH_<3>,(aq)+OH^<->,(aq))
- (NH^&minus_<2>,(aq)+H_2O,(l) ightleftharpoons NH_<3>,(aq)+OH^<&minus>,(aq))
- (S^<2&minus>,(aq)+H_2O,(l) ightleftharpoons HS^&minus,(aq)+OH^&minus,(aq))

- (HBr,(aq)+H_2O,(l) ightleftharpoons H_3O^+,(aq)+Br^&minus,(aq))
- (NaH,(s)+NH_<3>,(aq) ightleftharpoons H_<2>,(g)+NaNH_<2>,(s))
- (OCH^<&minus>_<3>,(aq)+NH_<3>,(aq) ightleftharpoons CH_<3>OH,(aq)+NH^&minus_<2>,(aq))
- (NH_<3>,(aq)+HCl,(aq) ightleftharpoons NH^<+>_<4>,(aq)+Cl^&minus,(aq))

- (Li_3N)
- (NaH)
- (KBr)
- (C_2H_5NH_3Cl)

- (LiCH_3)
- (MgCl_2)
- (K_2O)
- ((CH_3)_2NH_2^+Br^&minus)

### Conceptual Answer

6. Strong acids have the smaller (pK_a).

a. Equilibrium lies primarily to the right because (HBr) ((pK_a=-8.7)) is a stronger acid than (H_<3>O^<+>) ((pK_a=-1.7)) and (H_<2>O) ((pK_a=14)) is a stronger base than (Br^-) ( (pK_a=-8.7)).

b. Equilibrium lies primarily to the left because (H_<2>) ((pK_a=36)) is a stronger acid than (NH_<3>) ((pK_a=38)) and ((NaNH_2)) ((pK_a=38)) is a stronger base than (NaH) ((pK_a=35)).

c. Equilibrium lies primarily to the left because (CH_<3>OH) ((pK_a=17)) is a stronger acid than (NH_<3>) ((pK_a=38)) and (NH_<2>^<->) ((pK_a=38)) is a stronger base than (OCH_<3>^<->) ((pK_a=25)).

d. Equilibrium lies to the right because (HCl) ((pK_a=-7)) is a stronger acid than (NH_<4>^<+>) ((pK_a=9.3)) and (NH_<3>) is a stronger base than (Cl^<->) ((pK_a=-7)).

7. To identify the strongest base we can determine their weakest conjugate acid. The conjugate acids of (CH_<3>^<->), (NH_<2>^<->), and (S_<2>^<->) are (CH_<4>), (NH_<3>), and (HS^<->), respectively. Next, we consider that acidity increases with positive charge on the molecule, thus ruling out that (S_<2>^<->) is the weakest base. Finally, we consider that acidity increases with electronegativity, therefore (NH_<3>) is the second most basic and (CH_<4>) is the most basic. To distinguish between the strength of the acids (HIO_3), (H_<2>SO_<4>), and (HClO_4) we can consider that the higher electronegativity and oxidation state of the central nonmetal is the more acidic, therefore the order of acidity is: (HIO_3)<(H_<2>SO_<4>)<(HClO_4) because electronegativity and oxidation state increases as follows: (I(+5)<S(+6)<Cl(+7)).

8. It is not accurate to say that a 2.0 M solution of (H_2SO_4), which contains two acidic protons per molecule, is 4.0 M in (H^+) because a 2.0 M solution of (H_2SO_4) is equivalent to 4.0 N in (H^+).

9. Alkalinity is a measure of acid neutralizing capability. The basicity of the soil is defined this way because bases such as (HCO_<3>^<->) and (CO_<3>^<2->) can neutralize acids in soil. Because most soil has a pH between 6 and 8, alkalinity can be estimated by its carbonate species alone. At a near neutral pH, most carbonate species are bicarbonate.

10. Aqueous solutions of salts such as (CaCl_<2>) are neutral because it is created from hydrochloric acid (a strong acid) and calcium hydroxide (a strong base). An aqueous solution of (NaNH_2) is basic because it can deprotonate alkynes, alcohols, and a host of other functional groups with acidic protons such as esters and ketones.

a. (Li_3N) is a base because the lone pair on the nitrogen can accept a proton.

b. (NaH) is a base because the hydrogen has a negative charge.

c. (KBr) is neutral because it is formed from (HBr) (a strong acid) and (KOH) (a strong base).

d. (C_2H_5NH_3Cl) is acidic because it can donate a proton.

c. The pH is expected to remain the same. (K_<2>O,(aq)+H_2O,(l) ightleftharpoons 2,KOH,(aq))

13. (Sn(H_2O)_4^<2+>) is expected to be more acidic than (Pb(H_2O)_4^<2+>) because (Sn) is more electronegative than (Pb).

14. (Sn(H_2O)_6^<4+>) is expected to be more acidic than (Sn(H_2O)_4^<2+>) because the charge on (Sn) is greater ((4^+>2^+)).

15. Yes, it is possible the order of increasing base strength is: (LiH<NaH<RbH<CsH) because increasing base strength is dependent on decreasing electronegativity.

### Numerical Problems

- Arrange these acids in order of increasing strength.
- acid A: (pK_a = 1.52)
- acid B: (pK_a = 6.93)
- acid C: (pK_a = 3.86)

Given solutions with the same initial concentration of each acid, which would have the highest percent ionization?

- Arrange these bases in order of increasing strength:
- base A: (pK_b = 13.10)
- base B: (pK_b = 8.74)
- base C: (pK_b = 11.87)

Given solutions with the same initial concentration of each base, which would have the highest percent ionization?

- Calculate the (K_a) and the (pK_a) of the conjugate acid of a base with each (pK_b) value.
- 3.80
- 7.90
- 13.70
- 1.40
- &minus2.50

### Numerical Answers

1. Acids in order of increasing strength: (acid,B<acid,C<acid,A). Given the same initial concentration of each acid, the highest percent of ionization is acid A because it is the strongest acid.

2. Bases in order of increasing strength: (base,A<base,C<base,B). Given the solutions with the same initial concentration of each base, the higher percent of ionization is base A because it is the weakest base.

(pK_a+pK_b=14 ightarrow pK_a=14-pK_b=14-3.80=10.2)

(pK_a+pK_b=14 ightarrow pK_a=14-pK_b=14-7.90=6.10)

(pK_a+pK_b=14 ightarrow pK_a=14-pK_b=14-7.90=3.000 imes 10^<-1>)

(pK_a+pK_b=14 ightarrow pK_a=14-pK_b=14-1.40=12.6)

e. (pK_a+pK_b=14 ightarrow pK_a=14-pK_b=14-7.90=16.5)

(pK_a+pK_b=14 ightarrow pK_b=14-pK_a=14-4.20=9.80)

(pK_a+pK_b=14 ightarrow pK_a=14-pK_b=14-4.80=9.20)

## Problems & Exercises

#### 16.1: Hooke&rsquos Law: Stress and Strain Revisited

**19.**Fish are hung on a spring scale to determine their mass (most fishermen feel no obligation to truthfully report the mass).(a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load?

(b) What is the mass of a fish that stretches the spring 5.50 cm?

(c) How far apart are the half-kilogram marks on the scale?

**Solution**

(a) (displaystyle 1.23×10^3N/m)

(b) (displaystyle 6.88 kg)

(c) (displaystyle 4.00 mm)**20.**It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke&rsquos law and is depressed 0.75 cm by its maximum load of 120 kg.(a) What is the spring&rsquos effective spring constant?

(b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85 kg team?

**21.**One type of BB gun uses a spring-driven plunger to blow the BB from its barrel.(a) Calculate the force constant of its plunger&rsquos spring if you must compress it 0.150 m to drive the 0.0500-kg plunger to a top speed of 20.0 m/s.

(b) What force must be exerted to compress the spring?

**Solution**

(a) 889 N/m

(b) 133 N**22.**(a) The springs of a pickup truck act like a single spring with a force constant of (displaystyle 1.30×10^5N/m). By how much will the truck be depressed by its maximum load of 1000 kg?(b) If the pickup truck has four identical springs, what is the force constant of each?

**23.**When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m.(a) What is the force constant of the spring?

(b) Will the spring be compressed more when he hops down the road?

**Solution**

(a) (displaystyle 6.53×10^3N/m)

(b) Yes**24.**A spring has a length of 0.200 m when a 0.300-kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it.(a) What is the force constant of the spring?

(b) What is the unloaded length of the spring?

#### 16.2: Period and Frequency in Oscillations

**25.**What is the period of (displaystyle 60.0Hz) electrical power?**Solution**

16.7 ms**26.**If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?**Solution**

0.400 s/beats**27.**Find the frequency of a tuning fork that takes (displaystyle 2.50×10^<&minus3>s) to complete one oscillation.**Solution**

400 Hz**28.**A stroboscope is set to flash every (displaystyle 8.00×10^<&minus5>s). What is the frequency of the flashes?**Solution**

12,500 Hz**29.**A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at 30.0 m/s?**Solution**

1.50 kHz**30**.*Engineering Application*Each piston of an engine makes a sharp sound every other revolution of the engine.

(a) How fast is a race car going if its eight-cylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer?

(b) At how many revolutions per minute is the engine rotating?

**Solution**

(a) 93.8 m/s

(b) (displaystyle 11.3×10^3) rev/min#### 16.3: Simple Harmonic Motion: A Special Periodic Motion

**31.**A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?**Solution**

(displaystyle 2.37N/m)**32.**If the spring constant of a simple harmonic oscillator is doubled, by what factor will the mass of the system need to change in order for the frequency of the motion to remain the same?**33.**A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?**Solution**

0.389 kg**34.**By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?**35.**Suppose you attach the object with mass (displaystyle m) to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring&rsquos original rest length.(a) Show that the spring exerts an upward force of (displaystyle 2.00mg) on the object at its lowest point.

(b) If the spring has a force constant of (displaystyle 10.0N/m) and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations.

(c) Find the maximum velocity.

**36.**A diver on a diving board is undergoing simple harmonic motion. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver is a male whose period of simple harmonic oscillation is 1.05 s. What is his mass if the mass of the board is negligible?**Solution**

94.7 kg**37.**Suppose a diving board with no one on it bounces up and down in a simple harmonic motion with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the simple harmonic motion of a 75.0-kg diver on the board?**38.**The device pictured in Figure entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring constant.(a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its spring constant?

(b) What is the time for one complete bounce of this child? (c) What is the child&rsquos maximum velocity if the amplitude of her bounce is 0.200 m?

*This child&rsquos toy relies on springs to keep infants entertained. (credit: By Humboldthead, Flickr)***39.**A 90.0-kg skydiver hanging from a parachute bounces up and down with a period of 1.50 s. What is the new period of oscillation when a second skydiver, whose mass is 60.0 kg, hangs from the legs of the first, as seen in Figure.*The oscillations of one skydiver are about to be affected by a second skydiver. (credit: U.S. Army, www.army.mil)***Solution**

1.94 s#### 16.4: The Simple Pendulum

**As usual, the acceleration due to gravity in these problems is taken to be g=9.80m/s2, unless otherwise specified.****40.**What is the length of a pendulum that has a period of 0.500 s?**Solution**

6.21 cm**41.**Some people think a pendulum with a period of 1.00 s can be driven with &ldquomental energy&rdquo or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?**42.**What is the period of a 1.00-m-long pendulum?**Solution**

2.01 s**43.**How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?**44.**The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?**Solution**

2.23 Hz**45.**Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?**46.**(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is (displaystyle 9.79m/s^2) is moved to a location where it the acceleration due to gravity is (displaystyle 9.82m/s^2). What is its new period?(b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.

**Solution**

(a) 2.99541 s

(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by (displaystyle (0.01)^2=0.01%) so it is necessary to have at least 4 digits after the decimal to see the changes.**47.**A pendulum with a period of 2.00000 s in one location (displaystyle (g=9.80m/s^2)) is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?**48.**(a) What is the effect on the period of a pendulum if you double its length?(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?

**Solution**

(a) Period increases by a factor of 1.41 ((displaystyle sqrt<2>))

(b) Period decreases to 97.5% of old period**49.**Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is (displaystyle 1.63m/s^2).**50.**At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is (displaystyle 1.63m/s^2), if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock&rsquos hour hand to make one revolution on the Moon.**Solution**

Slow by a factor of 2.45**51.**Suppose the length of a clock&rsquos pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.**52.**If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?**Solution**

length must increase by 0.0116%.#### 16.5: Energy and the Simple Harmonic Oscillator

**53.**The length of nylon rope from which a mountain climber is suspended has a force constant of (displaystyle 1.40×10^4N/m).(a) What is the frequency at which he bounces, given his mass plus and the mass of his equipment are 90.0 kg?

(b) How much would this rope stretch to break the climber&rsquos fall if he free-falls 2.00 m before the rope runs out of slack? Hint: Use conservation of energy. Ignore the energy the climber gains as the rope stretches.

**Solution**

(a) (displaystyle 1.99 Hz)

(b) 50.2 cm**54.***Engineering Application*Near the top of the Citigroup Center building in New York City, there is an object with mass of (displaystyle 4.00×10^5)kg on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven&mdashthe driving force is transferred to the object, which oscillates instead of the entire building.

(a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s?

(b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?

**Solution**

(a) (displaystyle 3.95×10^6N/m)

(b) (displaystyle 7.90×10^6J)#### 16.6: Uniform Circular Motion and Simple Harmonic Motion

**55.**(a)What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of (displaystyle 1.50×10^6N/m), if the amplitude of the bounce is 0.200 cm?(b)What is the maximum energy stored in the spring?

**Solution**

a). 0.266 m/s

b). 3.00 J**56.**A novelty clock has a 0.0100-kg mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position?(b) How many joules of kinetic energy does the object have at its maximum velocity?

**57.**At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of (displaystyle x/X) give (displaystyle v=±v_/2), where (displaystyle X) is the amplitude of the motion? **Solution**

(displaystyle ±frac><2>) **58.**A ladybug sits 12.0 cm from the center of a Beatles music album spinning at 33.33 rpm. What is the maximum velocity of its shadow on the wall behind the turntable, if illuminated parallel to the record by the parallel rays of the setting Sun?#### 16.7: Damped Harmonic Motion

**59.**The amplitude of a lightly damped oscillator decreases by (displaystyle 3.0\%) during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?#### 16.8: Forced Oscillations and Resonance

**60.**How much energy must the shock absorbers of a 1200-kg car dissipate in order to damp a bounce that initially has a velocity of 0.800 m/s at the equilibrium position? Assume the car returns to its original vertical position.**Solution**

384 J**61.**If a car has a suspension system with a force constant of (displaystyle 5.00×10^4N/m), how much energy must the car&rsquos shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?**62.**(a) How much will a spring that has a force constant of 40.0 N/m be stretched by an object with a mass of 0.500 kg when hung motionless from the spring?(b) Calculate the decrease in gravitational potential energy of the 0.500-kg object when it descends this distance.

(c) Part of this gravitational energy goes into the spring. Calculate the energy stored in the spring by this stretch, and compare it with the gravitational potential energy. Explain where the rest of the energy might go.

**Solution**

(a). 0.123 m

(b). &minus0.600 J

(c). 0.300 J. The rest of the energy may go into heat caused by friction and other damping forces.**63.**Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction (displaystyle &mu_s=0.100).(a) How far can the spring be stretched without moving the mass?

(b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coefficient of friction is (displaystyle &mu_k=0.0850), what total distance does it travel before stopping? Assume it starts at the maximum amplitude.

**64.**Engineering Application: A suspension bridge oscillates with an effective force constant of (displaystyle 1.00×10^8N/m).(a) How much energy is needed to make it oscillate with an amplitude of 0.100 m?

(b) If soldiers march across the bridge with a cadence equal to the bridge&rsquos natural frequency and impart (displaystyle 1.00×10^4J) of energy each second, how long does it take for the bridge&rsquos oscillations to go from 0.100 m to 0.500 m amplitude?

**Solution**

(a) (displaystyle 5.00×10^5J)

(b) (displaystyle 1.20×10^3s)#### 16.9: Waves

**65.**Storms in the South Pacific can create waves that travel all the way to the California coast, which are 12,000 km away. How long does it take them if they travel at 15.0 m/s?**Solution**

t=9.26 d**66.**Waves on a swimming pool propagate at 0.750 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.0 s. How far away is the other end of the pool?**67.**Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?**Solution**

f=40.0 Hz**68.**How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?**69.**Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake it the bridge twice per second, what is the propagation speed of the waves?**Solution**

(displaystyle v_w=16.0 m/s)**70.**What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at 0.800 m/s?**71.**What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?**Solution**

&lambda=700 m**72.**Radio waves transmitted through space at (displaystyle 3.00×10^8m/s) by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?**73.**Your ear is capable of differentiating sounds that arrive at the ear just 1.00 ms apart. What is the minimum distance between two speakers that produce sounds that arrive at noticeably different times on a day when the speed of sound is 340 m/s?**Solution**

d=34.0 cm**74.**(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, they compare the arrival times of S- and P-waves, which travel at different speeds. Figure) If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined?(b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)

*A seismograph as described in above problem.(credit: Oleg Alexandrov)*#### 16.10: Superposition and Interference

**75.**A car has two horns, one emitting a frequency of 199 Hz and the other emitting a frequency of 203 Hz. What beat frequency do they produce?**Solution**

(displaystyle f=4 Hz)**76.**The middle-C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?**77.**Two tuning forks having frequencies of 460 and 464 Hz are struck simultaneously. What average frequency will you hear, and what will the beat frequency be?**Solution**

462 Hz,

4 Hz**78.**Twin jet engines on an airplane are producing an average sound frequency of 4100 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?**79.**A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again.(a) What is the speed of the wave?

(b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

**Solution**

(a) 3.33 m/s

(b) 1.25 Hz**80.**Three adjacent keys on a piano (F, F-sharp, and G) are struck simultaneously, producing frequencies of 349, 370, and 392 Hz. What beat frequencies are produced by this discordant combination?#### 16.11: Energy in Waves: Intensity

**81.***Medical Application*Ultrasound of intensity (displaystyle 1.50×10^2W/m^2) is produced by the rectangular head of a medical imaging device measuring 3.00 by 5.00 cm. What is its power output?

**Solution**

0.225 W**82.**The low-frequency speaker of a stereo set has a surface area of (displaystyle 0.05m^2) and produces 1W of acoustical power. What is the intensity at the speaker? If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity (displaystyle 0.1W/m^2)?**83.**To increase intensity of a wave by a factor of 50, by what factor should the amplitude be increased?**Solution**

7.07**84.***Engineering Application*A device called an insolation meter is used to measure the intensity of sunlight has an area of (displaystyle 100 cm^2) and registers 6.50 W. What is the intensity in (displaystyle W/m^2)?

**85.***Astronomy Application*Energy from the Sun arrives at the top of the Earth&rsquos atmosphere with an intensity of (displaystyle 1.30kW/m^2). How long does it take for (displaystyle 1.8×10^9J) to arrive on an area of (displaystyle 1.00m^2)?

**Solution**

16.0 d**86.**Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?**Solution**

2.50 kW**87.***Engineering Application*(a) A photovoltaic array of (solar cells) is 10.0% efficient in gathering solar energy and converting it to electricity. If the average intensity of sunlight on one day is (displaystyle 700W/m^2), what area should your array have to gather energy at the rate of 100 W?

(b) What is the maximum cost of the array if it must pay for itself in two years of operation averaging 10.0 hours per day? Assume that it earns money at the rate of 9.00 ¢ per kilowatt-hour.

**88.**A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally (displaystyle 2.00×10^<&ndash5>W/m^2), but is turned up until the amplitude increases by 30.0%, what is the new intensity?**Solution**

(displaystyle 3.38×10^<&ndash5>W/m^2)**89.***Medical Application*(a) What is the intensity in (displaystyle W/m^2) of a laser beam used to burn away cancerous tissue that, when 90.0% absorbed, puts 500 J of energy into a circular spot 2.00 mm in diameter in 4.00 s?

(b) Discuss how this intensity compares to the average intensity of sunlight (about (displaystyle 700W/m^2)) and the implications that would have if the laser beam entered your eye. Note how your answer depends on the time duration of the exposure.

## Passive Tense Review

Do you remember the difference between

**active**and**passive**sentences?### In an active sentence:

Someone or something is performing an action.

*"The cat chased the mouse."*### In a passive sentence:

Someone or something is having something done to them.

*"The mouse was chased by the cat."*In order to make a passive sentence you need:

- A form of the verb 'to be'
- A past tense verb

e.g.*"Thousands of toys***are bought**every Christmas."

If we want to say who performed the action then we must add '

**by**'.

e.g.*"Thousands of toys are bought every Christmas***by parents**."**Can you convert the following sentences into the passive voice?**1. Jennifer bought the cake.

2. Millions of people visit Cape Town every year.

3. Emily and Patrick ate my chocolate!

4. Sam painted a beautiful picture.

5. Kevin drives Kate to work every day.

6. Peter killed the rat.

7. Caroline eats two bars of chocolate daily.

8. The waves hit the ship.

9. Sophie cleans the kitchen.

10. Everyone watches the fireworks.

## Software

The phone runs Android 10, along with Motorola&rsquos updated My UX interface. Motorola uses a light hand for its software overlay, so for the most part the software here is stock Android.

My UX brings features like Moto Actions, which allow you to enable gestures to quickly activate commonly used features. For example, you can flip the phone over to silence it, or flick your wrist twice while holding it to turn on the flashlight.

Although Motorola has yet to announce plans for any future software updates, we anticipate the phone will receive Android 11 at some point after it comes out.

## Sony E 16mm f/2.8 SEL16F28

The Sony NEX system was announced in May 2010, with three lenses: the 18-55 and 18-200mm zoom lenses, and the subject of this review, the 16mm ƒ/2.8 pancake prime lens.

The 16mm ƒ/2.8 was designed for the new Sony ''E'' mount, with the image circle filling only the APS-C sensor size (it would vignette if used with a full-frame camera, and it's not clear if that would even be possible). The lens gives an effective field of view of 24mm when mounted on a NEX3 or NEX5, and with the addition of bayonet-mount adapters, the lens can be used as a 20mm fisheye lens or a 18mm ultrawide angle lens.

The lens takes 49mm filters, and while a lens hood does not ship with the lens, the hood from the 18-55mm lens is compatible. The lens is available now separately for approximately $250, or as part of a NEX camera kit.

**Sharpness**

The Sony E 16mm ƒ/2.8 pancake is fairly soft when used wide open at the ƒ/2.8 aperture: around 3-4 blur units on average, with a small sweet spot that is moderately sharper (about 1.5 blur units) in the center. Stopping down does improve results for sharpness, but not incredibly: by ƒ/4, it's 2-3 blur units, and it hits its maximum sharpness at ƒ/5.6-8, where the softer parts of the frame are about 2 blur units, while the center improves to about 1. Diffraction limiting begins to set in at ƒ/11, and the image is quite soft at f/22. (Oddly, even at f/22, the corners are noticeably softer than the center.)**Chromatic Aberration**

We noted a high level of chromatic aberration present in images shot with the 16mm ƒ/2.8 E, and not necessarily just in the corners. CA is most prominent when the lens is shot wide open at ƒ/2.8, and reduces slightly as the lens is stopped down.**Shading (''Vignetting'')**

Light falloff is a factor, but not a huge one, with this lens. When used wide open at ƒ/2.8, the corners of the image are 2/3 EV darker than the center when used at any other aperture, the corners are 1/3 EV darker than the center.**Distortion**

Distortion is usually a factor for wide-angle lenses, and so it's not surprise to find distortion in images shot with the 16mm ƒ/2.8 E. What is surprising is that it's a fairly prominent pincushion distortion - wide-angle lenses usually produce barrel distortion. In this case, there is some slight barrel distortion throughout the image (+0.2%), but in the corners, we note -0.7% distortion.**Autofocus Operation**

The Sony 16mm ƒ/2.8 E is quick to autofocus - the lens takes less than a second to slew through the entire range of focus. It is very fast as there is little to move inside the lens when focusing. Small changes in focus are conducted extremely quickly. The ring will turn all the way around with no stops. When using manual focus as you turn the ring the central section of the LCD displays an enlarged (7X or 14X) view of the image. It makes the camera very easy to use for manual focusing with this lens.**Macro**

Look elsewhere for macro performance - the lens has a magnification rating of just 0.073x. To its credit, the minimum close-focusing distance is short, at just 24cm (a little under a foot).**Build Quality and Handling**

A nice looking, very compact, pancake style lens that looks great on either the NEX3 or NEX5 bodies. The lens is very lightweight, weighing in at just 70 grams (2.5 oz). The front element is a tiny dime-size element, and the entire optical design of the lens is made up of just 5 elements, one of which is aspherical. The lens balances nicely on the NEX3 or 5 body, but it's worth noting that with the design of these cameras, the lens is offset to the left, and that does make that side of the camera slightly heavier.The only control surface on the lens is the focusing ring - there is simply no room on the lens for a depth-of-field scale or anything similar. All other functionality such as enabling or disabling autofocus is conducted on the camera. The rubber focusing ring is 3/8'' wide, and will rotate forever in either direction with no hard or soft stops.

While the lens doesn't ship with a lens hood, the hood from the 18-55mm lens will fit on this lens and function well (however, we don't know whether it's the recommended hood). When attached, the petal-shaped lens hood will add 1 1/8'' to the overall length of the lens, and could be reversed for storage on the bayonet mount.

The other accessories available for this lens are the wide-angle adapters, which we did not have an opportunity to test, but will allow the lens to operate as a 20mm fisheye lens (VCL-ECF1 adapter) or an 18mm ultrawide angle lens (VCL-ECU1 adapter).

**Alternatives**Right now, there's precious little in the way of alternatives for the Sony E mount. An adapter exists to allow you to mount standard alpha lenses, but the only comparable lens would be the 16mm ƒ/2.8 fisheye. The NEX kit lens, the 18-55mm ƒ/3.5-5.6, offers almost-as-wide performance, but without the constant ƒ/2.8 aperture.

**Conclusion**

The size of the lens make a NEX camera system*almost*pocketable. If you like the idea of a wide prime for your NEX camera, right now there's simply nothing else available. It's an interesting little lens, but its price point and its performance are perhaps limiting it to strictly consumer application.**Sample Photos**The VFA target should give you a good idea of sharpness in the center and corners, as well as some idea of the extent of barrel or pincushion distortion and chromatic aberration, while the Still Life subject may help in judging contrast and color. We shoot both images using the default JPEG settings and manual white balance of our test bodies, so the images should be quite consistent from lens to lens.

As appropriate, we shoot these with both full-frame and sub-frame bodies, at a range of focal lengths, and at both maximum aperture and ƒ/8. For the ''VFA'' target (the viewfinder accuracy target from Imaging Resource), we also provide sample crops from the center and upper-left corner of each shot, so you can quickly get a sense of relative sharpness, without having to download and inspect the full-res images. To avoid space limitations with the layout of our review pages, indexes to the test shots launch in separate windows.

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