Calculus Problem

Question

According to U.S. postal regulations, the girth plus the length of a parcel sent by mail may not exceed 108 inches, whereby "girth" mean the perimeter of the smallest end. What is the largest possible volume of a rectangular parcel with a square end that can be sent by mail? Such a package is shown below, with x and y measured in inches. Assume y > x What are the dimensions of the package of the largest volume?

Find a formula for the volume of the parcel in terms of x and y.

Volume= _________ cubic meters

The problem statement tells us that the parcel's girth plus length may not exceed 108 inches. in order to maximize volume, we assume that we will actually need the girth plus length to equal 108 inches. What equation does this produce involving x and y?

Equation: _____________

Solve this equation for y in terms of x.

y= ______________

Find a formula for the volume V (x) in terms of x.

V(x)=_______________

What is the domain of the function V? Note that x must be positive and y>x; consider how these facts, and the constraint that girth plus length is 108 inches, limit the possible values for x. Give your answer using interval notation.

Domain: ____________

Find the absolute maximum volume of the parcel on the domain you established above and hence also determine the dimensions of the box of greatest volume.

Maximum Volume:______________ cubic inches

Optimal Dimensions: x=___________ and y=____________ inches

Answer 1

V=yx²

108=4x+y

V=(108-4x)x²

since y>x ,,,,,108-4x>x,,,,0<x<108/5

dV/dx=-4x²+2x(108-4x)

set to zero

12x=216

x=18

y=108-72=36