maximum or minimum

MAXIMUM/MINIMUM PROBLEMS

The following problems are maximum/minimum optimization problems. They illustrate one of the most important applications of the first derivative. Many students find these problems intimidating because they are "word" problems, and because there does not appear to be a pattern to these problems. However, if you are patient you can minimize your anxiety and maximize your success with these problems by following these guidelines :

GUIDELINES FOR SOLVING MAX./MIN. PROBLEMS

1. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly.

2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts.

3. Define variables to be used and carefully label your picture or diagram with these variables. This step is very important because it leads directly or indirectly to the creation of mathematical equations.

4. Write down all equations which are related to your problem or diagram. Clearly denote that equation which you are asked to maximize or minimize. Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint" equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs. Some problems may have NO constraint equation. Some problems may have two or more constraint equations.

5. Before differentiating, make sure that the optimization equation is a function of only one variable. Then differentiate using the well-known rules of differentiation.

6. Verify that your result is a maximum or minimum value using the first or second derivative test for extrema.

The following problems range in difficulty from average to challenging.

PROBLEM 1: Find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.

PROBLEM 2: Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen?

PROBLEM 3: An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume?

PROBLEM 4: A container in the shape of a right circular cylinder with no top has surface area 3 ft.2 what height h and base radius r will maximize the volume of the cylinder?

PROBLEM 5: A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume?

PROBLEM 6: Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse.

PROBLEM 7 : Find the point (x, y) on the graph of nearest the point (4, 0).

PROBLEM 8 : A cylindrical can is to hold 20 m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

PROBLEM 9 : You are standing at the edge of a slow-moving river which is one mile wide and wish to return to your campground on the opposite side of the river. You can swim at 2 mph and walk at 3 mph. You must first swim across the river to any point on the opposite bank. From there walk to the campground, which is one mile from the point directly across the river from where you start your swim, what route will take the least amount of time?

PROBLEM 10: Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 12 feet, what dimensions will result in the rectangle having largest possible area?

PROBLEM 11: There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees?

PROBLEM 12 : Find the dimensions of the rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and graph of y=8-x3 . (See diagram.)

PROBLEM 13 : Consider a rectangle of perimeter 12 inches. Form a cylinder by revolving this rectangle about one of its edges. What dimensions of the rectangle will result in a cylinder of maximum volume ?

PROBLEM 14 : A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle of the movie screen is as large as possible ? (See diagram.)

PROBLEM 15 : Find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in a sphere of radius 2.

PROBLEM 16 : What angle between two edges of length 3 will result in an isosceles triangle with the largest area ? (See diagram.)

PROBLEM 17 : Of all lines tangent to the graph of , find the tangent lines of mimimum slope and maximum slope.

PROBLEM 18 : Find the length of the shortest ladder that will reach over an 8-ft. high fence to a large wall which is 3 ft. behind the fence. (See diagram.)

PROBLEM 19 : Find the point P = (x, 0) on the x-axis which minimizes the sum of the squares of the distances from P to (0, 0) and from P to (3, 2).

PROBLEM 20 : Car B is 30 miles directly east of Car A and begins moving west at 90 mph. At the same moment car A begins moving north at 60 mph. What will be the minimum distance between the cars and at what time t does the minimum distance occur ?

PROBLEM 21 : A rectangular piece of paper is 12 inches high and six inches wide. The lower right-hand corner is folded over so as to reach the leftmost edge of the paper (See diagram.).

Find the minimum length of the resulting crease.

SOLUTIONS:

SOLUTION 1 : Let variables x and y represent two nonnegative numbers. The sum of the two numbers is given to be

9 = x + y ,

so that

y = 9 - x .

We wish to MAXIMIZE the PRODUCT

P = x y2 .

However, before we differentiate the right-hand side, we will write it as a function of x only. Substitute for y getting

P = x y2

= x ( 9-x)2 .

Now differentiate this equation using the product rule and chain rule, getting

P' = x (2) ( 9-x)(-1) + (1) ( 9-x)2

= ( 9-x) [ -2x + ( 9-x) ]

= ( 9-x) [ 9-3x ]

= ( 9-x) (3)[ 3-x ]

= 0

for

x=9 or x=3 .

Note that since both x and y are nonnegative numbers and their sum is 9, it follows

that . See the adjoining sign chart for P' .

If

x=3 and y=6 ,

then

P= 108

is the largest possible product.

SOLUTION 2 : Let variable x be the width of the pen and variable y the length of the pen.

The total amount of fencing is given to be

500 = 5 (width) + 2 (length) = 5x + 2y ,

so that

2y = 500 - 5x

or

y = 250 - (5/2)x .

We wish to MAXIMIZE the total AREA of the pen

A = (width) (length) = x y .


A = x y

= x ( 250 - (5/2)x)

= 250x - (5/2)x2 .

Now differentiate this equation, getting

A' = 250 - (5/2) 2x

= 250 - 5x

= 5 (50 - x )

= 0

for

x=50 .

Note that since there are 5 lengths of x in this construction and 500 feet of fencing, it

follows that . See the adjoining sign chart for A' .

If

x=50 ft. and y=125 ft. ,

then

A = 6250 ft.2

is the largest possible area of the pen.

SOLUTION 3 : Let variable x be the length of one edge of the square base and variable y the height of the box.

The total surface area of the box is given to be

48 = (area of base) + 4 (area of one side) = x2 + 4 (xy) ,

so that

4xy = 48 - x2

or

.

We wish to MAXIMIZE the total VOLUME of the box

V = (length) (width) (height) = (x) (x) (y) = x2 y .


V = x2 y

= 12x - (1/4)x3 .


V' = 12 - (1/4)3x2

= 12 - (3/4)x2

= (3/4)(16 - x2 )

= (3/4)(4 - x)(4 + x)

= 0

for

x=4 or x=-4 .

But since variable x measures a distance and x > 0 . Since the base of the box

is square and there are 48 ft.2 of material, it follows that . See the adjoining sign chart for V' .

If

x=4 ft. and y=2 ft. ,

then

V = 32 ft.3

is the largest possible volume of the box.

SOLUTION 4 : Let variable r be the radius of the circular base and variable h the height of the cylinder.

The total surface area of the cylinder is given to be

(area of base) + (area of the curved side)

,

so that

or

.

We wish to MAXIMIZE the total VOLUME of the cylinder

V = (area of base) (height) .

However, before we differentiate the right-hand side, we will write it as a function of r only. Substitute for h getting

.


= 0

for

r=1 or r=-1 .

But since variable r measures a distance and r > 0 . Since the base of the box

is a circle and there are ft.2 of material, it follows that . See the adjoining sign chart for V' .

If

r=1 ft. and h=1 ft. ,

then

ft.3

is the largest possible volume of the cylinder.

SOLUTION 5 : Let variable x be the length of one edge of the square cut from each corner of the sheet of cardboard.

After removing the corners and folding up the flaps, we have an ordinary rectangular box.

We wish to MAXIMIZE the total VOLUME of the box

V = (length) (width) (height) = (4-2x) (3-2x) (x) .

Now differentiate this equation using the triple product rule, getting

V' = (-2) (3-2x) (x) + (4-2x) (-2) (x) + (4-2x) (3-2x) (1)

= -6x + 4x2 - 8x + 4x2 + 4x2 - 14x + 12

= 12x2 - 28x + 12

= 4 ( 3x2 - 7x + 3 )

= 0

for (Use the quadratic formula.)

,

i.e., for

or .

But since variable x measures a distance. In addition, the short edge of the

cardboard is 3 ft., so it follows that . See the adjoining sign chart for V' .

If

ft. ,

then

ft.3

is largest possible volume of the box.

SOLUTION 6 : Let variable x be the x-intercept and variable y the y-intercept of the line passing throught the point (8/9, 3) .

Set up a relationship between x and y using similar triangles.

One relationship is

,

so that

.

We wish to MINIMIZE the length of the HYPOTENUSE of the triangle

.


.

Now differentiate this equation using the chain rule and quotient rule, getting

(Factor a 2 out of the big brackets and simplify.)

= 0 ,

so that (If , then A=0 .)

.

By factoring out x , it follows that

,

so that (If AB= 0 , then A=0 or B=0 .)

x=0

(Impossible, since x> 8/9. Why ?) or

.

Then

,

so that

(x-8/9)3 = 8 ,

x-8/9 = 2 ,

and

x = 26/9 .

See the adjoining sign chart for H' .

If

x = 26/9 and y=13/3 ,

then

is the shortest possible hypotenuse.

SOLUTION 7 : Let (x, y) represent a randomly chosen point on the graph of .

We wish to MINIMIZE the DISTANCE between points (x, y) and (4, 0) ,

.


.

Now differentiate this equation using the chain rule, getting

= 0 ,


2x-7 = 0 ,

or

x =7/2 .

See the adjoining sign chart for L' .

If

x = 7/2 and ,

then

is the shortest possible distance from (4, 0) to the graph of .

SOLUTION 8 : Let variable r be the radius of the circular base and variable h the height of the cylinder.

The total volume of the cylinder is given to be

(area of base) (height) ,

so that

.

We wish to MINIMIZE the total COST of construction of the cylinder

C = (total cost of bottom) + (total cost of top) + (total cost of side)

= (unit cost of bottom)(area of bottom) + (unit cost of top)(area of top) + (unit cost of side) (area of side)

(For convenience drop the signs until the end of the problem.)

.


.


(Get a common denominator and combine fractions.)

= 0 ,


,

r3 = 8 ,

or

r = 2 .

Since variable r measures a distance, it must satisfy r > 0 . See the adjoining sign chart for C' .

If

r=2 m. and h=5 m. ,

then

is the least possible cost of construction.

SOLUTION 9 : Let variable x be the distance denoted in the given diagram.

Assume that you travel at the following rates :

SWIM : 2 mph

WALK : 3 mph .

Recall that if travel is at a CONSTANT rate of speed, then

(distance traveled) = (rate of travel) (time elapsed)

or

D = R T ,

so that time elapsed is

.

We wish to MINIMIZE the total TIME elapsed

T = (swim time) + (walk time)

= (swim distance)/(swim rate) + (walk distance)/(walk rate)

.


= 0 ,

so that

and

.

Square both sides of this equation, getting

9x2 = 4 (1 + x2) = 4 + 4x2 ,

so that

5x2 = 4 ,

x2 = 4/5 ,

or

.

But since variable x measures a distance and . See the adjoining sign chart for T' .

If

mi.

then

hr.

is the shortest possible time of travel.

SOLUTION 10 : Let variable x be the width and variable y the length of the rectangular portion of the window.

The semi-circular portion of the window has length

(radius) .

The perimeter (distance around outside only) of the window is given to be

so that

or

.

We wish to MAXIMIZE the total AREA of the RECTANGLE

A = (width) (length) = x y .


A = x y

.


= 0

for

,

i.e.,

.

Since variable x measures distance, . In addition, x is largest when y = 0 and the

window is in the shape of a semi-circle. Thus, (Why ?). See the adjoining sign chart for A' .

If

ft. and y=3 ft. ,

then

ft.2

is the largest possible area of the rectangle.

SOLUTION 11 : Let variable x be the ADDITIONAL trees planted in the existing orchard. We wish to MAXIMIZE the total PRODUCTION of apples

P = (number of trees) (apple output per tree)

= ( 50 + x ) ( 800 - 10x )

= 40,000 + 300 x - 10 x2 .


P' = 300 - 20 x

= 20 ( 15 - x )

= 0

for

x=15 .

See the adjoining sign chart for P' .

If

x = 15 additional trees ,

then

P = 42,250 apples

is the largest possible production of apples.

SOLUTION 12 : Let variable x be the length of the base and variable y the height of the inscribed rectangle.

We wish to MAXIMIZE the total AREA of the rectangle

A = (length of base) (height) = xy .


A = x y

= x ( 8 - x3 )

= 8x - x4 .


A' = 8 - 4 x3

= 4 ( 2 - x3 )

= 0 ,

so that

x3 = 2

and

.

Note that . See the adjoining sign chart for A' .

If

and y = 6 ,

then

is the largest possible area for the inscribed rectangle.

SOLUTION 13 : Let variable r be the length of the base and variable h the height of the rectangle.

It is given that the perimeter of the rectangle is

12 = 2r + 2h

so that

2h = 12 - 2r

and

h = 6 - r .

We wish to MAXIMIZE the total VOLUME of the resulting CYLINDER

V = (area of base) (height) .


.


= 0

for

r=0 or r=4 .

Since variable r measures distance and the perimeter of the rectangle is 12, . See the adjoining sign chart for V' .

If

r = 4 ft. and h = 2 ft. ,

then

ft. ft.3

is the largest possible volume for the cylinder.

SOLUTION 14 : Let variable be the viewing angle and variable x the distance as denoted in the diagram. We seek to write angle as a function of distance x . Introduce angle as in the diagram below.

It follows from basic trigonometry that

so that

(Equation 1)

.

In a similar fashion

so that

,

or

(Equation 2)

.

Use from Equation 1 to substitute into Equation 2, getting

.

We wish to MAXIMIZE angle THETA given in this equation. Differentiate this equation, getting

= 0 ,

so that

,

30 x2 + 3000 = 10 x2 + 9000 ,

20 x2 = 6000 ,

x2 = 300 ,

for

.

But since variable x measures distance and . If , then

(These are well-known values from basic trigonometry.)

radians

or

degrees .

See the adjoining sign chart for .

If

ft. ft.

then

degrees radians

is the largest possible viewing angle.

maximum or minimum

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