According to U.S. postal regulations, the girth plus the length of a parcel sent

Question

According to U.S. postal regulations, the girth plus the length of a parcel sent by mail may not exceed 108 inches, where by "girth" we 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. Assume y > x. What are the dimensions of the package of largest volume? Find a formula for the volume of the parcel in terms of x and y. Volume = 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 both x and y must be positive; consider how the constraint that girth plus length is 108 inches limits the possible values for x. Give your answer using interval notation. Domain: Find the absolute maximum of the volume of the parcel on the domain you established above and hence also determine the dimensions of the box of greatest volume. Maximum Volume = Optimal dimensions: x = and y =

Explanation / Answer

volume V=x2y

equation : 4x+y =108

y =108-4x

v(x)=x2(108-4x)=108x2-4x3

domain 0<x<108/4 =>0<x<27

for maximum volume dv/dx =0 ,d2v/dx2<0

dv/dx =216x -12x2=0

=>x(216-12x)=0

=>x=0,x=216/12=18

d2v/dx2=-24x

x=18,d2v/dx2=-24*18<0

so maximum volume when x =18in

y =108-4*18 =36in

maximum volume v=182*36=11664 cubic inches

optimal dimensions :x=18in ,y =36in

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