- 4.7.1. 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 Example 4.32, 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 4.62). Given ft of wire fencing, determine the dimensions that would create a garden of maximum area. What is the maximum area?
Let denote the length of the side of the garden perpendicular to the rock wall and denote the length of the side parallel to the rock wall. Then the area of the garden is
We want to find the maximum possible area subject to the constraint that the total fencing is From Figure 4.62, the total amount of fencing used will be Therefore, the constraint equation is
Solving this equation for we have Thus, we can write the area as
Before trying to maximize the area function we need to determine the domain under consideration. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. Therefore, we need and Since if then Therefore, we are trying to determine the maximum value of for over the open interval We do not know that a function necessarily has a maximum value over an open interval. However, we do know that a continuous function has an absolute maximum (and absolute minimum) over a closed interval. Therefore, let’s consider the function over the closed interval If the maximum value occurs at an interior point, then we have found the value in the open interval that maximizes the area of the garden. Therefore, we consider the following problem:
Maximize over the interval
As mentioned earlier, since is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. These extreme values occur either at endpoints or critical points. At the endpoints, Since the area is positive for all in the open interval the maximum must occur at a critical point. Differentiating the function we obtain
Therefore, the only critical point is (Figure 4.63). We conclude that the maximum area must occur when Then we have To maximize the area of the garden, let ft and The area of this garden is
Now let’s look at a general strategy for solving optimization problems similar to Example 4.32.
- Introduce all variables. If applicable, draw a figure and label all variables.
- 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).
- Write a formula for the quantity to be maximized or minimized in terms of the variables. This formula may involve more than one variable.
- Write any equations relating the independent variables in the formula from step Use these equations to write the quantity to be maximized or minimized as a function of one variable.
- Identify the domain of consideration for the function in step based on the physical problem to be solved.
- Locate the maximum or minimum value of the function from step 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 in. by 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?
Step 1: Let be the side length of the square to be removed from each corner (Figure 4.64). Then, the remaining four flaps can be folded up to form an open-top box. Let be the volume of the resulting box.
Step 2: We are trying to maximize the volume of a box. Therefore, the problem is to maximize
Step 3: As mentioned in step are trying to maximize the volume of a box. The volume of a box is where are the length, width, and height, respectively.
Step 4: From Figure 4.64, we see that the height of the box is inches, the length is inches, and the width is inches. Therefore, the volume of the box is
Step 5: To determine the domain of consideration, let’s examine Figure 4.64. Certainly, we need Furthermore, the side length of the square cannot be greater than or equal to half the length of the shorter side, in.; otherwise, one of the flaps would be completely cut off. Therefore, we are trying to determine whether there is a maximum volume of the box for over the open interval Since is a continuous function over the closed interval we know will have an absolute maximum over the closed interval. Therefore, we consider over the closed interval and check whether the absolute maximum occurs at an interior point.
Step 6: Since is a continuous function over the closed, bounded interval must have an absolute maximum (and an absolute minimum). Since at the endpoints and for the maximum must occur at a critical point. The derivative is
To find the critical points, we need to solve the equation
Dividing both sides of this equation by the problem simplifies to solving the equation
Using the quadratic formula, we find that the critical points are
Since is not in the domain of consideration, the only critical point we need to consider is Therefore, the volume is maximized if we let The maximum volume is as shown in the following graph.
Watch a video about optimizing the volume of a box.
Suppose the dimensions of the cardboard in Example 4.33 are 20 in. by 30 in. Let be the side length of each square and write the volume of the open-top box as a function of Determine the domain of consideration for
Minimizing Travel Time
An island is due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of and swims at a rate of How far should the visitor run before swimming to minimize the time it takes to reach the island?
Step 1: Let be the distance running and let be the distance swimming (Figure 4.66). Let be the time it takes to get from the cabin to the island.
Step 2: The problem is to minimize
Step 3: To find the time spent traveling from the cabin to the island, add the time spent running and the time spent swimming. Since Distance Rate Time the time spent running is
and the time spent swimming is
Therefore, the total time spent traveling is
Step 4: From Figure 4.66, the line segment of miles forms the hypotenuse of a right triangle with legs of length and Therefore, by the Pythagorean theorem, and we obtain Thus, the total time spent traveling is given by the function
Step 5: From Figure 4.66, we see that Therefore, is the domain of consideration.
Step 6: Since is a continuous function over a closed, bounded interval, it has a maximum and a minimum. Let’s begin by looking for any critical points of over the interval The derivative is
Squaring both sides of this equation, we see that if satisfies this equation, then must satisfy
We conclude that if is a critical point, then satisfies
Therefore, the possibilities for critical points are
Since is not in the domain, it is not a possibility for a critical point. On the other hand, is in the domain. Since we squared both sides of Equation 4.6 to arrive at the possible critical points, it remains to verify that satisfies Equation 4.6. Since does satisfy that equation, we conclude that is a critical point, and it is the only one. To justify that the time is minimized for this value of we just need to check the values of at the endpoints and and compare them with the value of at the critical point We find that and whereas Therefore, we conclude that has a local minimum at mi.
Suppose the island is mi from shore, and the distance from the cabin to the point on the shore closest to the island is Suppose a visitor swims at the rate of and runs at a rate of Let 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.
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.
Owners of a car rental company have determined that if they charge customers dollars per day to rent a car, where the number of cars they rent per day can be modeled by the linear function If they charge per day or less, they will rent all their cars. If they charge per day or more, they will not rent any cars. Assuming the owners plan to charge customers between $50 per day and per day to rent a car, how much should they charge to maximize their revenue?
Step 1: Let be the price charged per car per day and let be the number of cars rented per day. Let be the revenue per day.
Step 2: The problem is to maximize
Step 3: The revenue (per day) is equal to the number of cars rented per day times the price charged per car per day—that is,
Step 4: Since the number of cars rented per day is modeled by the linear function the revenue can be represented by the function
Step 5: Since the owners plan to charge between per car per day and per car per day, the problem is to find the maximum revenue for in the closed interval
Step 6: Since is a continuous function over the closed, bounded interval it has an absolute maximum (and an absolute minimum) in that interval. To find the maximum value, look for critical points. The derivative is Therefore, the critical point is When When When Therefore, the absolute maximum occurs at The car rental company should charge per day per car to maximize revenue as shown in the following figure.
A car rental company charges its customers dollars per day, where It has found that the number of cars rented per day can be modeled by the linear function How much should the company charge each customer to maximize revenue?
Maximizing the Area of an Inscribed Rectangle
A rectangle is to be inscribed in the ellipse
What should the dimensions of the rectangle be to maximize its area? What is the maximum area?
Step 1: For a rectangle to be inscribed in the ellipse, the sides of the rectangle must be parallel to the axes. Let be the length of the rectangle and be its width. Let be the area of the rectangle.
Step 2: The problem is to maximize
Step 3: The area of the rectangle is
Step 4: Let be the corner of the rectangle that lies in the first quadrant, as shown in Figure 4.68. We can write length and width Since and we have Therefore, the area is
Step 5: From Figure 4.68, we see that to inscribe a rectangle in the ellipse, the -coordinate of the corner in the first quadrant must satisfy Therefore, the problem reduces to looking for the maximum value of over the open interval Since will have an absolute maximum (and absolute minimum) over the closed interval we consider over the interval If the absolute maximum occurs at an interior point, then we have found an absolute maximum in the open interval.
Step 6: As mentioned earlier, is a continuous function over the closed, bounded interval Therefore, it has an absolute maximum (and absolute minimum). At the endpoints and For Therefore, the maximum must occur at a critical point. Taking the derivative of we obtain
To find critical points, we need to find where We can see that if is a solution of
then must satisfy
Therefore, Thus, are the possible solutions of Equation 4.7. Since we are considering over the interval is a possibility for a critical point, but is not. Therefore, we check whether is a solution of Equation 4.7. Since is a solution of Equation 4.7, we conclude that is the only critical point of in the interval Therefore, must have an absolute maximum at the critical point To determine the dimensions of the rectangle, we need to find the length and the width If then
Therefore, the dimensions of the rectangle are and The area of this rectangle is
Modify the area function if the rectangle is to be inscribed in the unit circle What is the domain of consideration?
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 over has an absolute minimum of at 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 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 in.3 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?
Step 1: Draw a rectangular box and introduce the variable to represent the length of each side of the square base; let represent the height of the box. Let denote the surface area of the open-top box.
Step 2: We need to minimize the surface area. Therefore, we need to minimize
Step 3: Since the box has an open top, we need only determine the area of the four vertical sides and the base. The area of each of the four vertical sides is The area of the base is Therefore, the surface area of the box is
Step 4: Since the volume of this box is and the volume is given as the constraint equation is
Solving the constraint equation for we have Therefore, we can write the surface area as a function of only:
Step 5: Since we are requiring that we cannot have Therefore, we need On the other hand, is allowed to have any positive value. Note that as becomes large, the height of the box becomes correspondingly small so that Similarly, as becomes small, the height of the box becomes correspondingly large. We conclude that the domain is the open, unbounded interval Note that, unlike the previous examples, we cannot reduce our problem to looking for an absolute maximum or absolute minimum over a closed, bounded interval. However, in the next step, we discover why this function must have an absolute minimum over the interval
Step 6: Note that as Also, as Since is a continuous function that approaches infinity at the ends, it must have an absolute minimum at some This minimum must occur at a critical point of The derivative is
Therefore, when Solving this equation for we obtain so Since this is the only critical point of the absolute minimum must occur at (see Figure 4.70). When Therefore, the dimensions of the box should be and With these dimensions, the surface area is
Consider the same open-top box, which is to have volume Suppose the cost of the material for the base is and the cost of the material for the sides is 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 be the side length of the base and be the height of the box.)
Section 4.7 Exercises
For the following exercises, answer by proof, counterexample, or explanation.
When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points?
Why do you need to check the endpoints for optimization problems?
True or False. For every continuous nonlinear function, you can find the value that maximizes the function.
True or False. For every continuous nonconstant function on a closed, finite domain, there exists at least one that minimizes or maximizes the function.
For the following exercises, set up and evaluate each optimization problem.
To carry a suitcase on an airplane, the length height of the box must be less than or equal to Assuming the height is fixed, show that the maximum volume is What height allows you to have the largest volume?
You are constructing a cardboard box with the dimensions 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?
Find the positive integer that minimizes the sum of the number and its reciprocal.
Find two positive integers such that their sum is and minimize and maximize the sum of their squares.
For the following exercises, consider the construction of a pen to enclose an area.
You have of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?
You have 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?
You need to construct a fence around an area of What are the dimensions of the rectangular pen to minimize the amount of material needed?
Two poles are connected by a wire that is also connected to the ground. The first pole is tall and the second pole is tall. There is a distance of between the two poles. Where should the wire be anchored to the ground to minimize the amount of wire needed?
[T] You are moving into a new apartment and notice there is a corner where the hallway narrows from What is the length of the longest item that can be carried horizontally around the corner?
A patient’s pulse measures To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression What value minimizes it?
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
You can run at a speed of mph and swim at a speed of mph and are located on the shore, miles east of an island that is mile north of the shoreline. How far should you run west to minimize the time needed to reach the island?
For the following problems, consider a lifeguard at a circular pool with diameter He must reach someone who is drowning on the exact opposite side of the pool, at position The lifeguard swims with a speed and runs around the pool at speed
Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle,
Find at what angle the lifeguard should swim to reach the drowning person in the least amount of time.
A truck uses gas as where represents the speed of the truck and represents the gallons of fuel per mile. At what speed is fuel consumption minimized?
For the following exercises, consider a limousine that gets at speed the chauffeur costs and gas is
Find the cost per mile at speed
Find the cheapest driving speed.
For the following exercises, consider a pizzeria that sell pizzas for a revenue of and costs where represents the number of pizzas.
Find the profit function for the number of pizzas. How many pizzas gives the largest profit per pizza?
Assume that and How many pizzas sold maximizes the profit?
Assume that and How many pizzas sold maximizes the profit?
For the following exercises, consider a wire long cut into two pieces. One piece forms a circle with radius and the other forms a square of side
Choose to maximize the sum of their areas.
Choose to minimize the sum of their areas.
For the following exercises, consider two nonnegative numbers and such that Maximize and minimize the quantities.
For the following exercises, draw the given optimization problem and solve.
Find the volume of the largest right circular cylinder that fits in a sphere of radius
Find the volume of the largest right cone that fits in a sphere of radius
Find the area of the largest rectangle that fits into the triangle with sides and
Find the largest volume of a cylinder that fits into a cone that has base radius and height
Find the dimensions of the closed cylinder volume that has the least amount of surface area.
Find the dimensions of a right cone with surface area that has the largest volume.
For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.
[T] Where is the line closest to the origin?
[T] Where is the line closest to point
[T] Where is the parabola closest to point
[T] Where is the parabola closest to point
For the following exercises, set up, but do not evaluate, each optimization problem.
A window is composed of a semicircle placed on top of a rectangle. If you have of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use to represent the radius of the semicircle.
You have a garden row of watermelon plants that produce an average of 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?
You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs and the material for the sides costs You need a box with volume Find the dimensions of the box that minimize cost. Use to represent the length of the side of the box.
You are building five identical pens adjacent to each other with a total area of as shown in the following figure. What dimensions should you use to minimize the amount of fencing?
You are the manager of an apartment complex with units. When you set rent at all apartments are rented. As you increase rent by one fewer apartment is rented. Maintenance costs run for each occupied unit. What is the rent that maximizes the total amount of profit?