4.6 Applied Optimization Problems

Learning Objectives

  • Set up and solve optimization problems in several applied fields.

One common application of calculus is calculating the minimum or maximum value of a function. For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter.

Solving Optimization Problems over a Closed, Bounded Interval

The basic idea of the optimization problems that follow is the same. We have a particular quantity that we are interested in maximizing or minimizing. However, we also have some auxiliary condition that needs to be satisfied. For example, in (Figure), we are interested in maximizing the area of a rectangular garden. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. However, what if we have some restriction on how much fencing we can use for the perimeter? In this case, we cannot make the garden as large as we like. Let’s look at how we can maximize the area of a rectangle subject to some constraint on the perimeter.

Maximizing the Area of a Garden

A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides ((Figure)). Given 100 ft of wire fencing, determine the dimensions that would create a garden of maximum area. What is the maximum area?

A drawing of a garden has x and y written on the vertical and horizontal sides, respectively. There is a rock wall running along the entire bottom horizontal length of the drawing.

Figure 1. We want to determine the measurements [latex]x[/latex] and [latex]y[/latex] that will create a garden with a maximum area using 100 ft of fencing.

Determine the maximum area if we want to make the same rectangular garden as in (Figure), but we have 200 ft of fencing.

Hint

We need to maximize the function [latex]A(x)=200x-2x^2[/latex] over the interval [latex][0,100][/latex].

Now let’s look at a general strategy for solving optimization problems similar to (Figure).

Problem-Solving Strategy: Solving Optimization Problems

  1. Introduce all variables. If applicable, draw a figure and label all variables.
  2. Determine which quantity is to be maximized or minimized, and for what range of values of the other variables (if this can be determined at this time).
  3. Write a formula for the quantity to be maximized or minimized in terms of the variables. This formula may involve more than one variable.
  4. Write any equations relating the independent variables in the formula from step 3. Use these equations to write the quantity to be maximized or minimized as a function of one variable.
  5. Identify the domain of consideration for the function in step 4 based on the physical problem to be solved.
  6. Locate the maximum or minimum value of the function from step 4. This step typically involves looking for critical points and evaluating a function at endpoints.

Now let’s apply this strategy to maximize the volume of an open-top box given a constraint on the amount of material to be used.

Maximizing the Volume of a Box

An open-top box is to be made from a 24 in. by 36 in. piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume?

Watch a video about optimizing the volume of a box.

Suppose the dimensions of the cardboard in (Figure) are 20 in. by 30 in. Let [latex]x[/latex] be the side length of each square and write the volume of the open-top box as a function of [latex]x[/latex]. Determine the domain of consideration for [latex]x[/latex].

Hint

The volume of the box is [latex]L \cdot W \cdot H[/latex].

Minimizing Travel Time

An island is [latex]2[/latex] mi due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is [latex]6[/latex] mi west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of [latex]8[/latex] mph and swims at a rate of [latex]3[/latex] mph. How far should the visitor run before swimming to minimize the time it takes to reach the island?

Suppose the island is 1 mi from shore, and the distance from the cabin to the point on the shore closest to the island is [latex]15[/latex] mi. Suppose a visitor swims at the rate of [latex]2.5[/latex] mph and runs at a rate of [latex]6[/latex] mph. Let [latex]x[/latex] denote the distance the visitor will run before swimming, and find a function for the time it takes the visitor to get from the cabin to the island.

Hint

The time [latex]T=T_{\text{running}}+T_{\text{swimming}}[/latex].

In business, companies are interested in maximizing revenue. In the following example, we consider a scenario in which a company has collected data on how many cars it is able to lease, depending on the price it charges its customers to rent a car. Let’s use these data to determine the price the company should charge to maximize the amount of money it brings in.

Maximizing Revenue

Owners of a car rental company have determined that if they charge customers [latex]p[/latex] dollars per day to rent a car, where [latex]50\le p\le 200[/latex], the number of cars [latex]n[/latex] they rent per day can be modeled by the linear function [latex]n(p)=1000-5p[/latex]. If they charge [latex]$50[/latex] per day or less, they will rent all their cars. If they charge [latex]$200[/latex] per day or more, they will not rent any cars. Assuming the owners plan to charge customers between [latex]$50[/latex] per day and [latex]$200[/latex] per day to rent a car, how much should they charge to maximize their revenue?

A car rental company charges its customers [latex]p[/latex] dollars per day, where [latex]60\le p\le 150[/latex]. It has found that the number of cars rented per day can be modeled by the linear function [latex]n(p)=750-5p[/latex]. How much should the company charge each customer to maximize revenue?

Hint

[latex]R(p)=n \times p[/latex], where [latex]n[/latex] is the number of cars rented and [latex]p[/latex] is the price charged per car.

Maximizing the Area of an Inscribed Rectangle

A rectangle is to be inscribed in the ellipse

[latex]\frac{x^2}{4}+y^2=1[/latex].

What should the dimensions of the rectangle be to maximize its area? What is the maximum area?

Modify the area function [latex]A[/latex] if the rectangle is to be inscribed in the unit circle [latex]x^2+y^2=1[/latex]. What is the domain of consideration?

Hint

If [latex](x,y)[/latex] is the vertex of the square that lies in the first quadrant, then the area of the square is [latex]A=(2x)(2y)=4xy[/latex].

Solving Optimization Problems when the Interval Is Not Closed or Is Unbounded

In the previous examples, we considered functions on closed, bounded domains. Consequently, by the extreme value theorem, we were guaranteed that the functions had absolute extrema. Let’s now consider functions for which the domain is neither closed nor bounded.

Many functions still have at least one absolute extrema, even if the domain is not closed or the domain is unbounded. For example, the function [latex]f(x)=x^2+4[/latex] over [latex](−\infty ,\infty )[/latex] has an absolute minimum of 4 at [latex]x=0[/latex]. Therefore, we can still consider functions over unbounded domains or open intervals and determine whether they have any absolute extrema. In the next example, we try to minimize a function over an unbounded domain. We will see that, although the domain of consideration is [latex](0,\infty )[/latex], the function has an absolute minimum.

In the following example, we look at constructing a box of least surface area with a prescribed volume. It is not difficult to show that for a closed-top box, by symmetry, among all boxes with a specified volume, a cube will have the smallest surface area. Consequently, we consider the modified problem of determining which open-topped box with a specified volume has the smallest surface area.

Minimizing Surface Area

A rectangular box with a square base, an open top, and a volume of [latex]216 \, \text{in}^3[/latex] is to be constructed. What should the dimensions of the box be to minimize the surface area of the box? What is the minimum surface area?

Consider the same open-top box, which is to have volume [latex]216 \, \text{in}^3[/latex]. Suppose the cost of the material for the base is [latex]$0.20 / \text{in}^2[/latex] and the cost of the material for the sides is [latex]$0.30 / \text{in}^2[/latex] and we are trying to minimize the cost of this box. Write the cost as a function of the side lengths of the base. (Let [latex]x[/latex] be the side length of the base and [latex]y[/latex] be the height of the box.)

Hint

If the cost of one of the sides is [latex]$0.30 / \text{in}^2[/latex], the cost of that side is [latex]0.30xy[/latex].

Key Concepts

  • To solve an optimization problem, begin by drawing a picture and introducing variables.
  • Find an equation relating the variables.
  • Find a function of one variable to describe the quantity that is to be minimized or maximized.
  • Look for critical points to locate local extrema.

For the following exercises, answer by proof, counterexample, or explanation.

1. When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points?

2. Why do you need to check the endpoints for optimization problems?

3. True or False. For every continuous nonlinear function, you can find the value [latex]x[/latex] that maximizes the function.

4. True or False. For every continuous nonconstant function on a closed, finite domain, there exists at least one [latex]x[/latex] that minimizes or maximizes the function.

For the following exercises, set up and evaluate each optimization problem.

5. To carry a suitcase on an airplane, the length + width + height of the box must be less than or equal to [latex]62[/latex] in. Assuming the height is fixed, show that the maximum volume is [latex]V=h(31-(\frac{1}{2})h)^2[/latex]. What height allows you to have the largest volume?

6. You are constructing a cardboard box with the dimensions 2 m by 4 m. You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions of the box with the largest volume?

A rectangle is drawn with height 2 and width 4. Each corner has a square with side length x marked on it.

7. Find the positive integer that minimizes the sum of the number and its reciprocal.

8. Find two positive integers such that their sum is 10, and minimize and maximize the sum of their squares.

For the following exercises, consider the construction of a pen to enclose an area.

9. You have [latex]400[/latex] ft of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?

10. You have [latex]800[/latex] ft of fencing to make a pen for hogs. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area?

11. You need to construct a fence around an area of [latex]1600[/latex] ft. What are the dimensions of the rectangular pen to minimize the amount of material needed?

12. Two poles are connected by a wire that is also connected to the ground. The first pole is [latex]20[/latex] ft tall and the second pole is [latex]10[/latex] ft tall. There is a distance of [latex]30[/latex] ft between the two poles. Where should the wire be anchored to the ground to minimize the amount of wire needed?

Two poles are shown, one that is 10 tall and the other is 20 tall. A right triangle is made with the shorter pole with other side length x. The distance between the two poles is 30.

13. [T] You are moving into a new apartment and notice there is a corner where the hallway narrows from 8 ft to 6 ft. What is the length of the longest item that can be carried horizontally around the corner?

An upside L-shaped figure is drawn with the _ part being 6 wide and the | part being 8 wide. There is a line drawn from the _ part to the | part that touches the near corner of the shape to form a hypotenuse for a right triangle the other sides being the the rest of the _ and | parts. This line is marked L.

14. A patient’s pulse measures 70 bpm, 80 bpm, then 120 bpm. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression [latex](x-70)^2+(x-80)^2+(x-120)^2[/latex]. What value minimizes it?

15. In the previous problem, assume the patient was nervous during the third measurement, so we only weight that value half as much as the others. What is the value that minimizes [latex](x-70)^2+(x-80)^2+\frac{1}{2}(x-120)^2[/latex]?

16. You can run at a speed of 6 mph and swim at a speed of 3 mph and are located on the shore, 4 miles east of an island that is 1 mile north of the shoreline. How far should you run west to minimize the time needed to reach the island?

A rectangle is drawn that has height 1 and length 4. In the lower right corner, it is marked “You” and in the upper left corner it is marked “Island.”

For the following problems, consider a lifeguard at a circular pool with diameter [latex]40[/latex] m. He must reach someone who is drowning on the exact opposite side of the pool, at position [latex]C[/latex]. The lifeguard swims with a speed [latex]v[/latex] and runs around the pool at speed [latex]w=3v[/latex].

A circle is drawn with points A and C on a diameter. There is a point B drawn on the circle such that angle BAC form an acute angle θ.

17. Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, [latex]\theta[/latex].

18. Find at what angle [latex]\theta[/latex] the lifeguard should swim to reach the drowning person in the least amount of time.

19. A truck uses gas at a rate of [latex]g(v)=av+\frac{b}{v}[/latex], where [latex]v[/latex] represents the speed of the truck and [latex]g[/latex] represents the gallons of fuel per mile. At what speed is fuel consumption minimized?

For the following exercises, consider a limousine that gets [latex]m(v)=\frac{(120-2v)}{5}[/latex] mi/gal at speed [latex]v[/latex], the chauffeur costs $15/h, and gas is $3.50/gal.

20. Find the cost per mile at speed [latex]v[/latex].

21. Find the cheapest driving speed.

For the following exercises, consider a pizzeria that sell pizzas for a revenue of [latex]R(x)=ax[/latex] and costs [latex]C(x)=b+cx+dx^2[/latex], where [latex]x[/latex] represents the number of pizzas.

22. Find the profit function for the number of pizzas. How many pizzas gives the largest profit per pizza?

23. Assume that [latex]R(x)=10x[/latex] and [latex]C(x)=2x+x^2[/latex]. How many pizzas sold maximizes the profit?

24. Assume that [latex]R(x)=15x[/latex], and [latex]C(x)=60+3x+\frac{1}{2}x^2[/latex]. How many pizzas sold maximizes the profit?

For the following exercises, consider a wire 4 ft long cut into two pieces. One piece forms a circle with radius [latex]r[/latex] and the other forms a square of side [latex]x[/latex].

25. Choose [latex]x[/latex] to maximize the sum of their areas.

26. Choose [latex]x[/latex] to minimize the sum of their areas.

For the following exercises, consider two nonnegative numbers [latex]x[/latex] and [latex]y[/latex] such that [latex]x+y=10[/latex]. Maximize and minimize the quantities.

27. [latex]xy[/latex]

28. [latex]x^2 y^2[/latex]

29. [latex]y-\frac{1}{x}[/latex]

30. [latex]x^2-y[/latex]

For the following exercises, draw the given optimization problem and solve.

31. Find the volume of the largest right circular cylinder that fits in a sphere of radius 1.

32. Find the volume of the largest right cone that fits in a sphere of radius 1.

33. Find the area of the largest rectangle that fits into the triangle with sides [latex]x=0, \, y=0[/latex] and [latex]\frac{x}{4}+\frac{y}{6}=1[/latex].

34. Find the largest volume of a cylinder that fits into a cone that has base radius [latex]R[/latex] and height [latex]h[/latex].

35. Find the dimensions of the closed cylinder volume [latex]V=16\pi [/latex] that has the least amount of surface area.

36. Find the dimensions of a right cone with surface area [latex]S=4\pi [/latex] that has the largest volume.

For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.

37. [T] Where is the line [latex]y=5-2x[/latex] closest to the origin?

38. [T] Where is the line [latex]y=5-2x[/latex] closest to point [latex](1,1)[/latex]?

39. [T] Where is the parabola [latex]y=x^2[/latex] closest to point [latex](2,0)[/latex]?

40. [T] Where is the parabola [latex]y=x^2[/latex] closest to point [latex](0,3)[/latex]?

For the following exercises, set up, but do not evaluate, each optimization problem.

41. A window is composed of a semicircle placed on top of a rectangle. If you have 20 ft of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use [latex]r[/latex] to represent the radius of the semicircle.

A semicircular window is drawn with radius r.

42. You have a garden row of 20 watermelon plants that produce an average of 30 watermelons apiece. For any additional watermelon plants planted, the output per watermelon plant drops by one watermelon. How many extra watermelon plants should you plant?

43. You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs [latex]$5 / \text{ft}^2[/latex] and the material for the sides costs [latex]$2 / \text{ft}^2[/latex]. You need a box with volume [latex]4 \, \text{ft}^2[/latex]. Find the dimensions of the box that minimize cost. Use [latex]x[/latex] to represent the length of the side of the box.

44. You are building five identical pens adjacent to each other with a total area of [latex]1000 \, \text{m}^2[/latex], as shown in the following figure. What dimensions should you use to minimize the amount of fencing?

A rectangle is divided into five sections, and each section has length y and width x.

45. You are the manager of an apartment complex with 50 units. When you set rent at $800/month, all apartments are rented. As you increase rent by $25/month, one fewer apartment is rented. Maintenance costs run $50/month for each occupied unit. What is the rent that maximizes the total amount of profit?

Glossary

optimization problems
problems that are solved by finding the maximum or minimum value of a function