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Calculus Volume 1

4.7 Applied Optimization Problems

Calculus Volume 14.7 Applied Optimization Problems
  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Key Terms
    13. Key Equations
    14. Key Concepts
    15. Chapter Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index

Learning Objectives

  • 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.

Example 4.32

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 100100 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 4.62 We want to determine the measurements xx and yy that will create a garden with a maximum area using 100100 ft of fencing.
Checkpoint 4.31

Determine the maximum area if we want to make the same rectangular garden as in Figure 4.63, but we have 200200 ft of fencing.

Now let’s look at a general strategy for solving optimization problems similar to Example 4.32.

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 3.3. 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 44 based on the physical problem to be solved.
  6. Locate the maximum or minimum value of the function from step 4.4. 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.

Example 4.33

Maximizing the Volume of a Box

An open-top box is to be made from a 2424 in. by 3636 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?

Media

Watch a video about optimizing the volume of a box.

Checkpoint 4.32

Suppose the dimensions of the cardboard in Example 4.33 are 20 in. by 30 in. Let xx be the side length of each square and write the volume of the open-top box as a function of x.x. Determine the domain of consideration for x.x.

Example 4.34

Minimizing Travel Time

An island is 2mi2mi due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is 6mi6mi west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 8mph8mph and swims at a rate of 3mph.3mph. How far should the visitor run before swimming to minimize the time it takes to reach the island?

Checkpoint 4.33

Suppose the island is 11 mi from shore, and the distance from the cabin to the point on the shore closest to the island is 15mi.15mi. Suppose a visitor swims at the rate of 2.5mph2.5mph and runs at a rate of 6mph.6mph. Let xx 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.

Example 4.35

Maximizing Revenue

Owners of a car rental company have determined that if they charge customers pp dollars per day to rent a car, where 50p200,50p200, the number of cars nn they rent per day can be modeled by the linear function n(p)=10005p.n(p)=10005p. If they charge $50$50 per day or less, they will rent all their cars. If they charge $200$200 per day or more, they will not rent any cars. Assuming the owners plan to charge customers between $50 per day and $200$200 per day to rent a car, how much should they charge to maximize their revenue?

Checkpoint 4.34

A car rental company charges its customers pp dollars per day, where 60p150.60p150. It has found that the number of cars rented per day can be modeled by the linear function n(p)=7505p.n(p)=7505p. How much should the company charge each customer to maximize revenue?

Example 4.36

Maximizing the Area of an Inscribed Rectangle

A rectangle is to be inscribed in the ellipse

x24+y2=1.x24+y2=1.

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

Checkpoint 4.35

Modify the area function AA if the rectangle is to be inscribed in the unit circle x2+y2=1.x2+y2=1. 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 f(x)=x2+4f(x)=x2+4 over (,)(,) has an absolute minimum of 44 at x=0.x=0. 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 (0,),(0,), 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.

Example 4.37

Minimizing Surface Area

A rectangular box with a square base, an open top, and a volume of 216216 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?

Checkpoint 4.36

Consider the same open-top box, which is to have volume 216in.3.216in.3. Suppose the cost of the material for the base is 20¢/in.220¢/in.2 and the cost of the material for the sides is 30¢/in.230¢/in.2 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 xx be the side length of the base and yy be the height of the box.)

Section 4.7 Exercises

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

311.

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

312.

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

313.

True or False. For every continuous nonlinear function, you can find the value xx that maximizes the function.

314.

True or False. For every continuous nonconstant function on a closed, finite domain, there exists at least one xx that minimizes or maximizes the function.

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

315.

To carry a suitcase on an airplane, the length +width++width+ height of the box must be less than or equal to 62in.62in. Assuming the height is fixed, show that the maximum volume is V=h(31(12)h)2.V=h(31(12)h)2. What height allows you to have the largest volume?

316.

You are constructing a cardboard box with the dimensions 2 m by 4 m.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.
317.

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

318.

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

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

319.

You have 400ft400ft of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?

320.

You have 800ft800ft 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?

321.

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

322.

Two poles are connected by a wire that is also connected to the ground. The first pole is 20ft20ft tall and the second pole is 10ft10ft tall. There is a distance of 30ft30ft 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.
323.

[T] You are moving into a new apartment and notice there is a corner where the hallway narrows from 8 ft to 6 ft.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.
324.

A patient’s pulse measures 70 bpm, 80 bpm, then 120 bpm.70 bpm, 80 bpm, then 120 bpm. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression (x70)2+(x80)2+(x120)2?(x70)2+(x80)2+(x120)2? What value minimizes it?

325.

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 (x70)2+(x80)2+12(x120)2?(x70)2+(x80)2+12(x120)2?

326.

You can run at a speed of 66 mph and swim at a speed of 33 mph and are located on the shore, 44 miles east of an island that is 11 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 40m.40m. He must reach someone who is drowning on the exact opposite side of the pool, at position C.C. The lifeguard swims with a speed vv and runs around the pool at speed w=3v.w=3v.

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.
327.

Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, θ.θ.

328.

Find at what angle θθ the lifeguard should swim to reach the drowning person in the least amount of time.

329.

A truck uses gas as g(v)=av+bv,g(v)=av+bv, where vv represents the speed of the truck and gg represents the gallons of fuel per mile. At what speed is fuel consumption minimized?

For the following exercises, consider a limousine that gets m(v)=(1202v)5mi/galm(v)=(1202v)5mi/gal at speed v,v, the chauffeur costs $15/h,$15/h, and gas is $3.5/gal.$3.5/gal.

330.

Find the cost per mile at speed v.v.

331.

Find the cheapest driving speed.

For the following exercises, consider a pizzeria that sell pizzas for a revenue of R(x)=axR(x)=ax and costs C(x)=b+cx+dx2,C(x)=b+cx+dx2, where xx represents the number of pizzas.

332.

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

333.

Assume that R(x)=10xR(x)=10x and C(x)=2x+x2.C(x)=2x+x2. How many pizzas sold maximizes the profit?

334.

Assume that R(x)=15x,R(x)=15x, and C(x)=60+3x+12x2.C(x)=60+3x+12x2. How many pizzas sold maximizes the profit?

For the following exercises, consider a wire 4ft4ft long cut into two pieces. One piece forms a circle with radius rr and the other forms a square of side x.x.

335.

Choose xx to maximize the sum of their areas.

336.

Choose xx to minimize the sum of their areas.

For the following exercises, consider two nonnegative numbers xx and yy such that x+y=10.x+y=10. Maximize and minimize the quantities.

337.

xyxy

338.

x2y2x2y2

339.

y1xy1x

340.

x2yx2y

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

341.

Find the volume of the largest right circular cylinder that fits in a sphere of radius 1.1.

342.

Find the volume of the largest right cone that fits in a sphere of radius 1.1.

343.

Find the area of the largest rectangle that fits into the triangle with sides x=0,y=0x=0,y=0 and x4+y6=1.x4+y6=1.

344.

Find the largest volume of a cylinder that fits into a cone that has base radius RR and height h.h.

345.

Find the dimensions of the closed cylinder volume V=16πV=16π that has the least amount of surface area.

346.

Find the dimensions of a right cone with surface area S=4πS=4π that has the largest volume.

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

347.

[T] Where is the line y=52xy=52x closest to the origin?

348.

[T] Where is the line y=52xy=52x closest to point (1,1)?(1,1)?

349.

[T] Where is the parabola y=x2y=x2 closest to point (2,0)?(2,0)?

350.

[T] Where is the parabola y=x2y=x2 closest to point (0,3)?(0,3)?

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

351.

A window is composed of a semicircle placed on top of a rectangle. If you have 20ft20ft of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use rr to represent the radius of the semicircle.

A semicircular window is drawn with radius r.
352.

You have a garden row of 2020 watermelon plants that produce an average of 3030 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?

353.

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $5/ft2$5/ft2 and the material for the sides costs $2/ft2.$2/ft2. You need a box with volume 4ft2.4ft2. Find the dimensions of the box that minimize cost. Use xx to represent the length of the side of the box.

354.

You are building five identical pens adjacent to each other with a total area of 1000m2,1000m2, 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.
355.

You are the manager of an apartment complex with 5050 units. When you set rent at $800/month,$800/month, all apartments are rented. As you increase rent by $25/month,$25/month, one fewer apartment is rented. Maintenance costs run $50/month$50/month for each occupied unit. What is the rent that maximizes the total amount of profit?

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