20 - 3 Extreme value
problems
This section deals with
relating maximum and minimum problems to your favorite type of problem.
Many business problems relate directly to maximizing profits while minimizing
expenses. This section deals with simple examples of these types
of problems.
Sample problems
1) What is the
area of the largest rectangular garden that a farmer can enclose with 500
feet of fencing if one side of the garden will be the side of a barn?
A(x) = L .
W
A(x) = x(500 - 2x) =
500x - 2x2
To find the maximum value,
find the derivative of A(x)
A'(x) = 500 - 4x
Set it equal to zero
and solve!
500 - 4x = 0
-4x = -500
x = 125
To have a maximum area
of the garden, the width must be 125' and the length must be 250'.
Therefore, the area is
maxed at (125)(250) = 31250 square feet!
2) A manufacturer
has 100 tons of metal that he can sell now with a profit of $5 a ton.
For each week that he delays shipment, he can produce another 10 tons of
metal. However, for each week he waits, the profit drops 25 cents
a ton. If he can sell the metal at any time, when is the best time
to sell so that his profit is maximized?
Let x = the number of
weeks to wait!
| Ship |
Amount of metal |
Profit per ton |
Total profit |
| now |
100 |
5 |
500 |
| in x weeks |
(100 + 10x) |
(5 - .25x) |
500 +25x -2.5x2 |
P(x) = 500 + 25x - 2.5x2
Find the derivative!
P'(x) = 25 - 5x
Set P'(x) = 0 and solve
25 - 5x = 0
-5x = -25
x = 5
Best time to sell is
in 5 weeks. He will have 150 tons and the price will be $3.75 for
a maximum profit of $562.50.
3) Given a 20-unit
square of sheet metal, find the dimensions of an open box ( no top) of
greatest volume that can be made by cutting congruent squares from the
corners.
Use the diagram above
for the metal sheet. The corners are being folded up to form the
box. V(x) = L . W . H
V(x) = x(20 - 2x)2
= 400x - 80x2 + 4x3
Find the derivative to
find the maximum point!
V'(x) = 400 - 160x +
12x2
= 4(100 - 40x + 3x2)
= 4(10 - 3x)(10 - x)
We have two critical
points: x = 10 and x = 10/3
What happens when x =
10! The volume will be zero because you are cutting the metal in
half. You won't be able to form a box and the volume is zero.
This is a minimum value. Thus the maximum value happens at
x = 10/3
The volume = 10/3(20
- 20/3)2 = 16,000/27
cubic units!
Really hate word problems
don't you!!
