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.
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 [link], 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.
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 ([link]). 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 [link], 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
([link]). 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
Determine the maximum area if we want to make the same rectangular garden as in [link], but we have
ft of fencing.
The maximum area is
We need to maximize the function
over the interval
Now let’s look at a general strategy for solving optimization problems similar to [link].
Use these equations to write the quantity to be maximized or minimized as a function of one variable.
based on the physical problem to be solved.
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.
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 ([link]). 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 [link], 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 [link]. 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 [link] 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
The domain is
The volume of the box is
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 ([link]). 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 [link], 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 [link], 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
If
then
Therefore,
Squaring both sides of this equation, we see that if
satisfies this equation, then
must satisfy
which implies
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 [link] to arrive at the possible critical points, it remains to verify that
satisfies [link]. 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.
The time
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?
The company should charge
per car per day.
where
is the number of cars rented and
is the price charged per car.
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 [link]. We can write length
and width
Since
and
we have
Therefore, the area is
Step 5: From [link], 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 [link]. 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 [link]. Since
is a solution of [link], 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?
The domain of consideration is
If
is the vertex of the square that lies in the first quadrant, then the area of the square is
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.
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:
Therefore,
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 [link]). 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.)
dollars
If the cost of one of the sides is
the cost of that side is
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?
The critical points can be the minima, maxima, or neither.
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.
False;
has a minimum only
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?
in.
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.
approximately
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.
Maximal:
minimal:
and
Maximal:
minimal: none
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?
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