20 - 3 Extreme value problems
 
 
 
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!
 
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!!