Calculus Volume 1 · 4.7 Applied Optimization Problems

Applied optimization problems

23 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

How to approach these problems

Start with a picture and decide exactly which quantity is being maximized or minimized. Write it as a function, then use the constraint (a fixed perimeter, volume or budget) to get it down to one variable, and note the interval that variable can take. Find the critical points by setting the derivative to zero, then compare them with the endpoints of a closed interval; on an open interval, the sign of the derivative around the point, or the second derivative, tells you whether it is a maximum or a minimum. The usual mistakes: optimizing the constraint instead of the objective, skipping the endpoints, and answering with x when the question asks for the dimensions or the cost.

Exercises 311
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?
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Exercises 313
For the following exercises, answer by proof, counterexample, or explanation.
True or False. For every continuous nonlinear function, you can find the value $x$ that maximizes the function.
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Exercises 315
For the following exercises, set up and evaluate each optimization problem.
To carry a suitcase on an airplane, the length $+ width +$ height of the box must be less than or equal to $62 in .$ Assuming the base of the suitcase is square, show that the volume is $V = h (31 - (\frac{1}{2}) h)^{2} .$ What height allows you to have the largest volume?
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Exercises 317
For the following exercises, set up and evaluate each optimization problem.
Find the positive integer that minimizes the sum of the number and its reciprocal.
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Exercises 319
For the following exercises, consider the construction of a pen to enclose an area.
You have $400 ft$ of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?
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Exercises 321
For the following exercises, consider the construction of a pen to enclose an area.
You need to construct a fence around an area of $1600 ft^{2} .$ What are the dimensions of the rectangular pen to minimize the amount of material needed?
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Exercises 323
For the following exercises, consider the construction of a pen to enclose an area.
[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?
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Exercises 325
For the following exercises, consider the construction of a pen to enclose an area.
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 $(x - 70)^{2} + (x - 80)^{2} + \frac{1}{2} (x - 120)^{2} ?$
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Exercises 327
For the following problems, consider a lifeguard at a circular pool with diameter $40 m .$ He must reach someone who is drowning on the exact opposite side of the pool, at position $C .$ The lifeguard swims with a speed $v$ and runs around the pool at speed $w = 3 v .$
Find a function that measures the total amount of time it takes to reach the drowning person as a function of the swim angle, $θ .$
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Exercises 329
For the following problems, consider a lifeguard at a circular pool with diameter $40 m .$ He must reach someone who is drowning on the exact opposite side of the pool, at position $C .$ The lifeguard swims with a speed $v$ and runs around the pool at speed $w = 3 v .$
A truck uses gas as $g (v) = a v + \frac{b}{v},$ where represents the speed of the truck and represents the gallons of fuel per mile. Assuming and are positive, at what speed is fuel consumption minimized?
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Exercises 331
For the following exercises, consider a limousine that gets $m (v) = \frac{( 120 - 2 v )}{5} mi/gal$ at speed $v,$ the chauffeur costs $$15/h,$ and gas is $$ 3.5 / gal .$
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Exercises 333
For the following exercises, consider a pizzeria that sell pizzas for a revenue of $R (x) = a x$ and costs $C (x) = b + c x + d x^{2},$ where $x$ represents the number of pizzas $; a > c$ .
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Exercises 335
For the following exercises, consider a wire $4 ft$ long cut into two pieces. One piece forms a circle with radius $r$ and the other forms a square of side $x .$
Choose $x$ to maximize the sum of their areas.
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Exercises 337
For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x + y = 10.$ Maximize and minimize the quantities.
$x y$
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Exercises 339
For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x + y = 10.$ Maximize and minimize the quantities.
$y - \frac{1}{x}$
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Exercises 341
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 $1.$
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Exercises 343
For the following exercises, draw the given optimization problem and solve.
Find the area of the largest rectangle that fits into the triangle with sides $x = 0, y = 0$ and $\frac{x}{4} + \frac{y}{6} = 1.$
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Exercises 345
For the following exercises, draw the given optimization problem and solve.
Find the dimensions of the closed cylinder volume $V = 16 π$ that has the least amount of surface area.
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Exercises 347
For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.
[T] Where is the line $y = 5 - 2 x$ closest to the origin?
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Exercises 349
For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.
[T] Where is the parabola $y = x^{2}$ closest to point $(2, 0)?$
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Exercises 351
For the following exercises, write an equation that models each related-rates situation. Do not solve.
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 $r$ to represent the radius of the semicircle.
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Exercises 353
For the following exercises, write an equation that models each related-rates situation. Do not solve.
You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $$ 5 / ft^{2}$ and the material for the sides costs $$ 2 / ft^{2} .$ You need a box with volume $4 ft^{3} .$ Find the dimensions of the box that minimize cost. Use $x$ to represent the length of the side of the box.
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Exercises 355
For the following exercises, write an equation that models each related-rates situation. Do not solve.
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?
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Key terms · 1

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

Learn 4.7 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 4.7 in the textbook · This section on openstax.org

Applied optimization problems

23 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

How to approach these problems

Exercises 311

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?

Exercises 313

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

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

Exercises 315

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

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

Exercises 317

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

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

Exercises 319

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

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

Exercises 321

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

You need to construct a fence around an area of $1600 ft^{2} .$ What are the dimensions of the rectangular pen to minimize the amount of material needed?

Exercises 323

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

[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.

Exercises 325

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

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 $(x - 70)^{2} + (x - 80)^{2} + \frac{1}{2} (x - 120)^{2} ?$

Exercises 327

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

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.

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

Exercises 329

A truck uses gas as $g (v) = a v + \frac{b}{v},$ where represents the speed of the truck and represents the gallons of fuel per mile. Assuming and are positive, at what speed is fuel consumption minimized?

Exercises 331

For the following exercises, consider a limousine that gets $m (v) = \frac{( 120 - 2 v )}{5} mi/gal$ at speed $v,$ the chauffeur costs $$15/h,$ and gas is $$ 3.5 / gal .$

Exercises 333

For the following exercises, consider a pizzeria that sell pizzas for a revenue of $R (x) = a x$ and costs $C (x) = b + c x + d x^{2},$ where $x$ represents the number of pizzas $; a > c$ .

Exercises 335

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

Choose $x$ to maximize the sum of their areas.

Exercises 337

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

$x y$

Exercises 339

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

$y - \frac{1}{x}$

Exercises 341

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 $1.$

Exercises 343

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

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

Exercises 345

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

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

Exercises 347

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

[T] Where is the line $y = 5 - 2 x$ closest to the origin?

Exercises 349

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

[T] Where is the parabola $y = x^{2}$ closest to point $(2, 0)?$

Exercises 351

For the following exercises, write an equation that models each related-rates situation. Do not solve.

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 $r$ to represent the radius of the semicircle.

A semicircular window is drawn with radius r.

Exercises 353

For the following exercises, write an equation that models each related-rates situation. Do not solve.

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $$ 5 / ft^{2}$ and the material for the sides costs $$ 2 / ft^{2} .$ You need a box with volume $4 ft^{3} .$ Find the dimensions of the box that minimize cost. Use $x$ to represent the length of the side of the box.

Exercises 355

For the following exercises, write an equation that models each related-rates situation. Do not solve.

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?

Key terms · 1

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

Learn 4.7 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 4.7 in the textbook · This section on openstax.org