# Calculus - Optimization

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UBC parcel post regulations states that packages must have length plus girth of no more than 84 inches. Find the dimension of the cylindrical package of greatest volume that is mailable by parcel post. What is the greatest volume? Make a sketch to indicate your variables.

• Calculus - Optimization -

Let
L=length, then
R=(84-L)/2

Volume, V = πR²L
Express V in terms of L using R=(84-L)/2

Equate dV/dL=0 and solve for L.

• Calculus - Optimization -

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A parcel service will only accept packages with a length of no more than 96 inches and length plus girth of no more than 156 inches. (See the figure below.) Assuming that the front face of the package (as shown in the figure) is square, what is the largest volume package that the parcel service will accept?

• Calculus - Optimization -

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