Page 109 - How To Solve Word Problems In Calculus

Step5 (Optional. If you are convinced that the box must
have a maximum volume, stop here.)

V (x) =−3x V (5) < 0
V has a relative maximum at 5. Since it is the only relative
extremum for positive x, the absolute maximum volume is
3
125 in .

EXAMPLE 12
3
A cylindrical can is to contain 2000 in of liquid. What di-
mensions will minimize the amount of metal used in the con-
struction of the can?

Solution
Step1
Let r and h represent the radius and height, respectively,
of the can.

r

h

Step2
S = 2πrh + 2πr 2

Step3
3
Since the volume of the can must be 2000 in ,

2
πr h = 2000
2000
h =
πr 2
96