Applied Max and Min Problems

Objective: To use the methods of this chapter to solve applied

optimization problems.

Classification

• The applied optimization problems that we will consider in this section fall into the following two categories:

1) Problems that reduce to maximizing or minimizing a continuous function over a finite closed interval.

2) Problems that reduce to maximizing or minimizing a continuous function over an infinite interval or a finite interval that is not closed.

Example 1

• A garden is to be laid out in a rectangular area and protected by a chicken wire fence. What is the largest possible area of the garden if only 100 running feet of chicken wire is available?

Example 1

• A garden is to be laid out in a rectangular area and protected by a chicken wire fence. What is the largest possible area of the garden if only 100 running feet of chicken wire is available?

• x = length of the rectangle (ft)• y = width of the rectangle (ft)• A = area of the rectangle (ft2)• The domain is 0 < x < 50

Example 1

• x = length of the rectangle (ft)• y = width of the rectangle (ft)• A = area of the rectangle (ft2) P = perimeter of fence• We are maximizing the area, so we need an equation

for the area in one variable. We will use two separate equations and substitute to get an area equation in one variable.

Example 1

• x = length of the rectangle (ft)• y = width of the rectangle (ft)• A = area of the rectangle (ft2) P = perimeter of fence• We are maximizing the area, so we need an equation

for the area in one variable. We will use two separate equations and substitute to get an area equation in one variable.

A = xy P = 2x + 2y; 100 = 2x + 2y; y = 50 - x

Example 1

• x = length of the rectangle (ft)• y = width of the rectangle (ft)• A = area of the rectangle (ft2) P = perimeter of fence• A = xy P = 2x + 2y; 100 = 2x + 2y; y = 50 – x A = x(50 – x) = 50x – x2

dA/dx = 50 – 2x ___+___|___-___ 25 max

Example 1

___+___|___-___ 25 max• We must now check the endpoints as well as the

critical numbers.• f(0) = 0• f(25) = 625• f(50) = 0

Example 1

___+___|___-___ 25 max• We must now check the endpoints as well as the

critical numbers.• f(0) = 0• f(25) = 625 Maximum area of 625 ft2 @ x = 25 ft• f(50) = 0

Example 2

• An open box is to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain a box with the largest volume?

• The largest the squares can be is 8 in (why?) so the domain is 0 < x < 8

Example 2

• An open box is to be made from a 16-inch by 30-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain a box with the largest volume?

• We are maximizing the volume, so we need an equation for volume. V = x(30 – 2x)(16 – 2x)

Example 2

• We are maximizing the volume, so we need an equation for volume. V = x(30 – 2x)(16 – 2x)

• V = 480x – 92x2 + 4x3

• dV/dx = 480 – 184x + 12x2

= 4(x – 12)(3x – 10)

Example 2

• We are maximizing the volume, so we need an equation for volume. V = x(30 – 2x)(16 – 2x)

• V = 480x – 92x2 + 4x3

• dV/dx = 480 – 184x + 12x2

= 4(x – 12)(3x – 10)• The only critical number in the domain is 10/3 so:• f(0) = 0• f(10/3) = 726• f(8) = 0

Example 2

• We are maximizing the volume, so we need an equation for volume. V = x(30 – 2x)(16 – 2x)

• V = 480x – 92x2 + 4x3

• dV/dx = 480 – 184x + 12x2

= 4(x – 12)(3x – 10)• The only critical number in the domain is 10/3 so:• f(0) = 0• f(10/3) = 726 Max volume of 726 in3 @ x = 10/3 in• f(8) = 0

Example 3

• An offshore oil well is located at a point W that is 5km from the closest point A on a straight shoreline. Oil is to be piped from W to a shore point B that is 8km from A by piping it on a straight line under water from W to some shore point P between A & B and then onto B via pipe along the shoreline. If the cost of laying pipe is $1,000,000/km under water and $500,000/km over land, where should the point P be located to minimize the cost of laying the pipe?

Example 3

• x = the distance (in km) between A and P• c = cost (in millions) for the entire pipeline• Using the Pythagorean Theorem, the distance from

W to P is • The distance from P to B is (8 – x)• We are minimizing cost, so we need a cost equation.

252 x

)8(2

1251 2 xxc

Example 3

• Now, we take the derivative to find the critical numbers.

)8(2

1251 2 xxc

2

1

252

x

x

dx

dc

2

1

250

2

x

x

Example 3

• Now, we take the derivative to find the critical numbers.

)8(2

1251 2 xxc

2

1

252

x

x

dx

dc

2

1

252

x

x

252 2 xx

254 22 xx

3

252 x3

5x2

1

250

2

x

x

Example 3

• We now need to check the endpoints and the critical number to find the min.

• The domain is 0 < x < 8 so:• F(0) = 9• F(8) = 9.433

• Min cost of 8.33 million @ x = km 33.83

5

f 3

5

Example 4

• Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 in and height 10 in.

Example 4

• Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 in and height 10 in.

• r = radius of cylinder• h = height of cylinder• V = volume

hrV 2

Example 4

• Again, we need to eliminate one of the variables. We will do this with similar triangles.

hrV 2

6

1010

r

h

rh3

510

322

3

510

3

510 rrrrV

Example 4

• Now, we take the derivative and find the critical points.

32

3

510 rrV 2520 rr

dr

dV

)4(5 rrdr

dV

Example 4

• The domain of the radius is 0 < r < 6, so we check the endpoints and the critical number.

• f(0) = 0• f(4) = 167.55 Max volume of 167.55 in3 @ x = 4 in• f(6) = 0

)4(5 rrdr

dV

Example 5

• A closed cylindrical can is to hold 1000 cm3 of liquid. How should we choose the height and radius to minimize the amount of material needed to manufacture the can?

• There are no physical restraints on this problem, so this is not a finite interval. We need to confirm that our critical number is a minimum.

Example 5

• A closed cylindrical can is to hold 1000 cm3 of liquid. How should we choose the height and radius to minimize the amount of material needed to manufacture the can?

• h = height (cm) of can• r = radius (cm) of can• S = surface area (cm2) of can

Example 5

• Again, we will find two equations and substitute to get the surface area equation in one variable.

• h = height (cm) of can• r = radius (cm) of can• S = surface area (cm2) of can

rhrS 22 2

hr 21000

2

1000

rh

Example 5

• Again, we will find two equations and substitute to get the surface area equation in one variable.

• h = height (cm) of can• r = radius (cm) of can• S = surface area (cm2) of can

rhrS 22 2

hr 21000

2

1000

rh

22 100022

rrrS

rrS

20002 2

Example 5

• Now we find the derivative and check the critical number.

rrS

20002 2

2

20004

rr

dr

dS

2

200040

rr

Example 5

• Now we find the derivative and check the critical number.

rrS

20002 2

2

20004

rr

dr

dS

420003 r

cmr 4.52

200040

rr

2

20004

rr

20004 3 r

Example 5

• Now we find the derivative and check the critical number.

____-___|___+___ 5.4 min

rrS

20002 2 2

20004

rr

dr

dS

420003 r cmr 4.5

Example 6

• Find a point on the curve y = x2 that is closest to the point (18, 0).

Example 6

• Find a point on the curve y = x2 that is closest to the point (18, 0).

• The distance L from (18, 0) and an arbitrary point

(x, y) on the curve is:• Since y = x2, we can say that

22 )0()18( yxL

42)18( xxL

Example 6

• Find a point on the curve y = x2 that is closest to the point (18, 0).

• The distance L from (18, 0) and an arbitrary point

(x, y) on the curve is:• Since y = x2, we can say that

• We are told that the distance L and the square of the distance L2 are minimized at the same value.

22 )0()18( yxL

42)18( xxL

Example 6

• We will now take the derivative of L2 to find any critical points

42)18( xxL

3624 3 xxdx

dS

32436242 xxxLS

Example 6

• We will now take the derivative of L2 to find any critical points

• This will factor to (x – 2)(2x2 + 4x + 9) [trust me!] ___-___|___+___ 2 min

42)18( xxL

3624 3 xxdx

dS

32436242 xxxLS

Example 6

• We will now take the derivative of L2 to find any critical points

• This will factor to (x – 2)(2x2 + 4x + 9) [trust me!] ___-___|___+___ The closest point is (2, 4) 2 min

42)18( xxL

3624 3 xxdx

dS

32436242 xxxLS

Homework

• Section 4.5

• 1, 3, 5, 11,19, 21, 23