Question 1

A company wants to design a rectangular garden with a fixed perimeter of $$100$$ meters. What dimensions will maximize the area of the garden?

Accepted Answer

$$x = 25 m$$, $$y = 25 m$$

Question 2

A farmer has $$2400 ft of $$fencing to create a rectangular field divided into three equal smaller rectangles by two fences parallel to one side. What should the dimensions of the field be to maximize the area?

Accepted Answer

$$x = 200 ft$$, $$y = 300 ft$$

Question 3

A box with a square base and open top must have a volume of $$32 m$^3. $$What are the dimensions of the box that minimize the amount of material used?

Accepted Answer

$$x = 4 m$$, $$y = 2 m$$

Question 4

Suppose an airline requires that the sum of the length, width, and height of a box-shaped piece of luggage does not exceed $$72$$ inches. What are the dimensions of a square-based box with the largest possible volume under this constraint?

Accepted Answer

$$x = 24 in$$, $$y = 24 in$$

Question 5

Find the positive numbers $$x$$ and $$y$$ such that $$x y = 12$$ and the sum $$2x + y is $$minimized.

Accepted Answer

$$x = \sqrt{6}$$, $$y = 2\sqrt{6}$$

Question 6

A manufacturer wants to build a box with a square base and open top using $$32 m$^3 of $$material. What is the minimum surface area required?

Accepted Answer

$$48 m$^2$$

Question 7

A rectangular field is to be fenced off along a river, so that no fence is needed along the river. If $$600$$ meters of fencing are available, what dimensions will maximize the area?

Accepted Answer

$$x = 150 m$$, $$y = 150 m$$

Question 8

A company wants to design a cylindrical can with a fixed volume of $$1000 cm$^3 to $$minimize the amount of material used. Which equation should be set up to find the optimal radius $$r$$?

Accepted Answer

Question 9

A rectangular piece of cardboard measuring $$30 cm by$$ $$40 cm is to be $$made into an open box by cutting equal squares from each corner and folding up the sides. What size should the squares be cut to maximize the volume of the box?

Accepted Answer

$$x = 4 cm$$

Question 10

A company wants to minimize the cost of producing a rectangular box with a square base and open top, given a fixed volume. Which of the following is the correct constraint equation for the volume?

Accepted Answer

$$V = x^2$ y$$

Question 11

A rectangular field is to be divided into three equal smaller rectangles by two fences parallel to one side. If the total fencing available is $$2400 ft$$, what is the equation relating $$x$$ and $$y ($$the sides of the field)?

Accepted Answer

$$6x + 4y = 2400$$

Question 12

Which of the following best describes the general approach to solving optimization problems in calculus?

Accepted Answer

Express the quantity to be optimized as a function of one variable, use constraints to eliminate other variables, find critical points by setting the derivative to zero, and test endpoints if necessary.

Question 13

A rectangular garden is to be constructed using $$100$$ meters of fencing. If one side of the garden is against a wall and does not require fencing, what are the dimensions that maximize the area?

Accepted Answer

$$x = 50 m$$, $$y = 25 m$$

Question 14

A box with a square base and open top is to have a volume of $$32 m$^3. $$What is the formula for the surface area $$A in $$terms of $$x ($$the length of the base)?

Accepted Answer

$$A = x^2$ + \frac{128}{x^2$}$$

Question 15

If $$x$$ and $$y$$ are positive numbers such that $$x y = 12$$, what is the minimum value of $$2x + y$$?

Accepted Answer

$$4\sqrt{6}$$

Question 16

A rectangular field is to be divided into three equal smaller rectangles by two fences parallel to one side. If the total fencing available is $$2400 ft$$, what is the maximum area that can be enclosed?

Accepted Answer

Question 17

Suppose you are designing a box with a square base and open top to maximize volume, given a fixed surface area. Which variable should you express in terms of the other to set up the optimization problem?

Accepted Answer

Express $$y in $$terms of $$x$$

Question 18

A box with a square base and open top must have a volume of $$32 m$^3. $$What is the value of $$x ($$the base length) that minimizes the surface area?

Accepted Answer

$$x = 4$$

Question 19

A rectangular field is to be divided into three equal smaller rectangles by two fences parallel to one side. If the total fencing available is $$2400 ft$$, what is the value of $$y ($$the length of the field) that maximizes the area?

Accepted Answer

$$y = 300 ft$$

Optimization Problems in Calculus: Applications of the Derivative 4.5

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