Optimization Practice Problems – Pike Page 1 of 15

Optimization Practice Problems

1. A company needs to design a cylindrical can that holds 475 cubic centimeters of fluid. What are the

dimensions of the can that would require the least amount of material?

2. A farmer has 500 feet of fence for constructing a rectangular corral. One side of the corral will be formed

by the barn and requires no fence. Three exterior fences and two interior fences partition the corral into

three rectangular pens. What are the dimensions of the corral that maximize the enclosed area? What is the

area of one of the three pens?

3. A rectangle is constructed with its base on the diameter of a semicircle with radius of 6 feet and its two

other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

4. An airline policy states that all baggage must be box-shaped with a sum of the length, width, and height

not exceeding 120 inches. What are the dimensions and volume of a square-based box with the greatest

volume under these conditions?

5. What point on the line 3x – y = 4 is closest to the point A(–2, 3)?

6. We want to construct a box whose base length is 3 times the base width. The material used to build the top

and bottom cost $11 per square foot and the material used to build the sides cost $7 per square foot. If the

box must have a volume of 75 cubic feet, determine the dimensions that will minimize the cost to build

the box.

7. At which points on the curve 3 5y 1 40x 3x does the tangent line have the largest slope?

8. The top and bottom margins of a poster are each 5 centimeters and the side margins are each 3

centimeters. If the area for printed material on the poster is fixed at 360 square centimeters, find the

dimensions of the poster with the smallest area.

9. Find the area of the largest trapezoid that can be inscribed in a circle with a radius of 5 inches and whose

base is a diameter of the circle.

10. A boat in the ocean is 3 miles from the nearest point on a straight shoreline; that point is 5 miles from a

restaurant on the shore. A person plans to row a boat to a point, P, on the shore and then walk along the

shore to the restaurant. If the person walks at 4 miles per hour and rows at 3 miles per hour, at which point

on the shore should the person land to minimize the total travel time?

Optimization Practice Problems – Pike Page 2 of 15

r radius

h height

Solutions

1. A company needs to design a cylindrical can that holds 475 cubic centimeters of fluid. What are the

dimensions of the can that would require the least amount of material?

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

2V r h 475

List the objective function (the function to be optimized).

top/bottm outside

2

2

Surface Area(A) A A

A 2( r ) 2 rh

A 2 r 2 rh

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

22

475r h 475 h

r

2

2

2

2

2 1

A 2 r 2 rh

475A 2 r 2 r

r

950A(r) 2 r

r

A(r) 2 r 950r

Use calculus to find the maximum or minimum value of the objective function.

To minimize A(r), we need to take the derivative of A(r) and find the critical value(s).

Remember to find the critical value(s), we set the derivative equal to zero.

2

2

2

3

3

A (r) 4 r 950r

950A (r) 4 r

r

9504 r 0

r

4 r 950 0

475r

2

Optimization Practice Problems – Pike Page 3 of 15

x = width of one pen

y = height of one pen

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

3475

x2

is a minimum because the derivative changes

from negative to zero to positive.

Answer the question.

13

3

3 2

223

475 475r 4.23

2 2

475 475 1900h 8.46

r 475

2

The dimensions that will minimize the material used are when then the radius of the can is

3475

r 4.232

centimeters and height of the can is 3 21900

h 8.46

centimeters.

2. A farmer has 500 feet of fence for constructing a rectangular corral. One side of the corral will be formed

by the barn and requires no fence. Three exterior fences and two interior fences partition the corral into

three rectangular pens. What are the dimensions of the corral that maximize the enclosed area? What is the

area of one of the three pens?

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

500 feet of fence to construct the corral, so 3x + 4y = 500

List the objective function (the function to be optimized).

A b h

A 3xy

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

500 3x

3x 4y 500 y4

2

A 3xy

500 3xA 3x

4

9A(x) x 375x

4

Optimization Practice Problems – Pike Page 4 of 15

Length 2x

Width y

Use calculus to find the maximum or minimum value of the objective function.

To minimize A(x), we need to take the derivative of A(x) and find the critical value(s).

Remember to find the critical value(s), we set the derivative equal to zero.

9A (x) x 375

2

9x 375 0

2

250x

3

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

250

x3

is a maximum because the derivative changes

from positive to zero to negative.

Answer the question.

250x

3

250500 3

500 3x 1253y

3 4 2

250 125A 3xy 3 15625

3 2

The maximum enclosed area 15265 square feet.

The dimensions of one of the pens is 250

3feet by

125

2feet.

3. A rectangle is constructed with its base on the diameter of a semicircle with radius of 6 feet and its two

other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

Rectangle is inscribed in a semicircle, so 2 2x y 36

List the objective function (the function to be optimized).

A L W

A 2xy

Optimization Practice Problems – Pike Page 5 of 15

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

2 2 2x y 36 y 36 x

2

A 2xy

A(x) 2x 36 x

Use calculus to find the maximum or minimum value of the objective function.

Maximize A(x) and find the critical numbers.

1

2 22

22

2

2 2

2

2

2

2

2

2

1A (x) 2x 36 x ( 2x) 36 x (2)

2

2xA (x) 2 36 x

36 x

2x 2(36 x )A (x)

36 x

4x 72A (x)

36 x

4x 720

36 x

4x 72 0

x 3 2

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

x 3 2 is a maximum because the derivative changes from positive to zero to negative.

Answer the question.

2

2

x 3 2

y 36 x 36 3 2 18

y 3 2

The dimensions that will produce the largest area are

Length 2x Width y

Length 6 2 feet Width 3 2 feet

Optimization Practice Problems – Pike Page 6 of 15

Length x

Width x

Height y

Note: x = 0 is impossible because the width cannot be zero.

4. An airline policy states that all baggage must be box-shaped with a sum of the length, width, and height

not exceeding 120 inches. What are the dimensions and volume of a square-based box with the greatest

volume under these conditions?

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

L W H 120

2x y 120

List the objective function (the function to be optimized).

2

V L W H

V x y

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

2x y 120 y 120 2x

2

2

3 2

V x y

V(x) x (120 2x)

V(x) 2x 120x

Use calculus to find the maximum or minimum value of the objective function.

Maximize V(x) and find the critical numbers.

2

2

V (x) 6x 240x

6x 240x 0

6x(x 40) 0

x 0 and x 40

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

x 40 is a maximum because the derivative changes from positive to zero to negative.

Answer the question.

x 40

y 120 2(40) 40

The dimensions that will produce the largest area are length = 40 inches, width = 40 inches, and

height = 40 inches.

Optimization Practice Problems – Pike Page 7 of 15

P(x, y) is a point on the line 3x y 4

5. What point on the line 3x – y = 4 is closest to the point A(–2, 3)?

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

Point P is a point on the line y = 3x – 4

List the objective function (the function to be optimized).

Point P needs to be as close to point A as possible, need to minimize the distance.

2 22 1 2 1d (x x ) (y y )

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

3x y 4 y 3x 4

2 2

22

2 2

2

d (x 2) (y 3) Plug in the point A( 2,3)

d(x) (x 2) (3x 4) 3

d(x) x 4x 4 9x 42x 49

d(x) 10x 38x 53

Use calculus to find the maximum or minimum value of the objective function.

Minimize d(x) and find the critical numbers.

1

2 2

2

2

1d (x) 10x 38x 53 (20x 38)

2

10x 19d (x)

10x 38x 53

10x 190

10x 38x 53

10x 19 0

19x

10

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

19

x10

is a minimum because the derivative changes from

negative to zero to positive.

Optimization Practice Problems – Pike Page 8 of 15

Length 3x

Width x

Height y

Answer the question.

19x

10

19y 3 4

10

17y

10

The point closest to point A(–3, 2) is 19 17

, .10 10

6. We want to construct a box whose base length is 3 times the base width. The material used to build the top

and bottom cost $11 per square foot and the material used to build the sides cost $7 per square foot. If the

box must have a volume of 75 cubic feet, determine the dimensions that will minimize the cost to build

the box.

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

The volume of the box must be 75 cubic feet.

2

V L W H 75

V 3x x y 75

3x y 75

List the objective function (the function to be optimized).

The cost of the box needs to be minimized.

top/bottom left /right front /back

2

2

C C C C

C 11 2 3x 7 2 3xy 7 2 xy

C 66x 56xy

Optimization Practice Problems – Pike Page 9 of 15

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

22

253x y 75 y

x

2

2

2

2

2 1

C 66x 56xy

25C(x) 66x 56x

x

1400C(x) 66x

x

C(x) 66x 1400x

Use calculus to find the maximum or minimum value of the objective function.

Minimize d(x) and find the critical numbers.

2

2

2

3

3

3

C (x) 132x 1400x

1400C (c) 132x

x

1400132x 0

x

132x 1400 0

350 381150x

33 33

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

3 381150

x33

is a minimum because the derivative

changes from negative to zero to positive.

Answer the question.

3

2 13 3

381150x

33

381150 381150C 66 1400

33 33

C 955.80

The least expensive cost of the 75 cubic foot box would be $955.80.

Optimization Practice Problems – Pike Page 10 of 15

7. At which points on the curve 3 5y 1 40x 3x does the tangent line have the largest slope?

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

Points must be on the curve 3 5y 1 40x 3x

List the objective function (the function to be optimized).

The slope of the tangent line need to be maximized.

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

3 5 2 4 2 4y 1 40x 3x y 120x 15x slope 12x 15x

Note: In order to maximize the slope at a point, we need to maximize the equation

2 4slope 120x 15x . In other words, we need to take the derivative of the slope.

Use calculus to find the maximum or minimum value of the objective function.

Maximize the slope and find the critical numbers.

2 4

3

3

2

slope 120x 15x

slope 240x 60x

60x 240x 0

60x(x 4) 0

x 0 or x 2

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

x = –2 and x = 2 are maximums because the

derivative changes from positive to zero to negative.

Answer the question.

2 4

2 4

x 2 slope 120( 2) 15( 2) 240

x 2 slope 120(2) 15(2) 240

The largest slope of 240 occurs at the points (–2, –223) and (2, 225).

Optimization Practice Problems – Pike Page 11 of 15

x length of printed area

y = width of printed area

8. The top and bottom margins of a poster are each 5 centimeters and the side margins are each 3

centimeters. If the area for printed material on the poster is fixed at 360 square centimeters, find the

dimensions of the poster with the smallest area.

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

The printed area must be 360 square centimeters.

xy 360

List the objective function (the function to be optimized).

The area of the poster needs to be as small as possible.

A L W

A (x 6)(y 10)

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

360

xy 360 yx

1

A (x 6)(y 10)

360A(x) (x 6) 10

x

2160A(x) 360 10x 60

x

2160A(x) 420 10x

x

A(x) 420 10x 2160x

6 6

Use calculus to find the maximum or minimum value of the objective function.

2

2

2

2

2

A (x) 10 2160x

2160A (x) 10

x

216010 0

x

10x 2160 0

x 216

x 6 6

x 6 6 is impossible because the length of the printed area can’t be negative.

Optimization Practice Problems – Pike Page 12 of 15

x width of one of the bases of the trapezoid

y height of the trapezoid

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

x 6 6 is a maximum because the derivative changes

from positive to zero to negative.

Answer the question.

360x 6 6 y 10 6

6 6

Length x 6 6 6 6

Width y 10 10 6 10

The dimensions of the poster with the smallest area would be 6 6 6 cm by 10 6 10 cm.

9. Find the area of the largest trapezoid that can be inscribed in a circle with a radius of 5 inches and whose

base is a diameter of the circle.

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

Trapezoid is inscribed in a circle, 2 2x y 25

List the objective function (the function to be optimized).

The area of the trapezoid needs to be as large as possible.

1 2h(b b )

A2

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

2 2 2x y 25 y 25 x

1 2

2

h(b b )A

2

y(2x 10)A y(x 5)

2

A(x) 25 x x 5

Note: 2b 10, the diameter of the circle.

Optimization Practice Problems – Pike Page 13 of 15

Use calculus to find the maximum or minimum value of the objective function.

1

2 2 2

2

2

2 2

2

2

2

2

2

2

2

1A (x) 25 x 1 (x 5) 25 x ( 2x)

2

x(x 5)A (x) 25 x

25 x

(25 x ) (x 5x)A (x)

25 x

2x 5x 25A (x)

25 x

2x 5x 250

25 x

2x 5x 25 0

5 ( 5) 4( 2)(25) 5 225 5 15x

2( 2) 4 4

5x 5 and

2

x = –5 is impossible because the base can’t be negative.

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

5

x2

is a maximum because the derivative changes from

positive to zero to negative.

Answer the question.

2

5x

2

5 5 25 15A 25 5 25

2 2 4 2

75 15 5 3 15A

4 2 2 2

75 3A

4

A 32.48

The largest possible area is 75 3

32.484

square centimeters.

Optimization Practice Problems – Pike Page 14 of 15

Walking Distance = 5 x

Rowing Distance = z

10. A boat in the ocean is 3 miles from the nearest point on a straight shoreline; that point is 5 miles from a

restaurant on the shore. A person plans to row a boat to a point, P, on the shore and then walk along the

shore to the restaurant. If the person walks at 4 miles per hour and rows at 3 miles per hour, at which point

on the shore should the person land to minimize the total travel time?

Define the variables used in the problem and organize the information using a picture.

List the constraint(s).

miWalk 4

hr

miRow 3

hr

List the objective function (the function to be optimized).

Total travel time needs to be minimized, distance d

time trate r

Write the objective function in terms of one variable. Use the constraint(s) to eliminate all but one

variable of the objective function.

2 2 2 2x 3 z z x 9

walk rowtotal walk row

walk row

total

2

d dt t t

r r

5 x zt

4 3

5 x x 9t(x)

4 3

Use calculus to find the maximum or minimum value of the objective function.

1

2 2

2

2

2

2

2 2 2

1x 9 2x

1 1 x2t (x)4 3 4 3 x 9

1 x0

4 3 x 9

x 14x 3 x 9

43 x 9

16x 9(x 9) 7x 81

9 9 7x

77

Optimization Practice Problems – Pike Page 15 of 15

Use the First Derivative Test to determine if each critical value is a maximum, minimum, or neither.

9 7

x7

is a minimum because the derivative changes

from negative to zero to positive.

Answer the question.

2

9 7x

7

9 7 819 7 35 9 79 95 7 77 7t4 3 4 3

1281 63

35 9 7 35 9 7 35 9 7 4 77 7t

28 3 28 3 28 7

35 9 7 16 7 35 7 7t

28 28

t 1.91

The least amount of time it will take to reach the restaurant is 35 7 7

1.9128

hours.