# calculus

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The USPS will not mail a package if the sum of its length and its girth (measuring around the package) exceed 108 inches. Determine:
(a) The volume of the package as a function of its radius r.

(b) For which radius r is the volume of the package a maximum?

(c) What is the maximum volume of the package?

• calculus -

h = length
r = radius of cylinder package (I assume)

girth = 2 pi r

h + 2pi r = 108 for maximum
so h = (108-2pir)

V = pi r^2 h
V = pi r^2 (108-2 pi r)
dh/dr = pi r^2 (-2pi) + (108-2 pi r)2pi r
for max
2 pi r^2 = 216r - 4 pi r^2
6 pi r^2 = 216 r
r = 216/(6 pi)
r = 11.5
then h = 108 -2 pi r = 36
then V = pi r^2 h = 14851 in^3

• calculus -

• calculus -

The question never specifies the shape of the package so it has really confused me on how to solve it.

• calculus -

Well by giving you r, it kind of implies a right circular cylinder.

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