A parcel delivery service will accept boxes for delivery provided the sum of the

Question

A parcel delivery service will accept boxes for delivery provided the sum of the length, width, and height not exceed 120 inches. Suppose you wish to ship a box whose height is twice its width and has the maximum possible volume.
a)Let x represent the width, in inches, of the box. Write a function V(x) that gives the volume of the box in terms of x.
b) Determine all critical points for V(x) using calculus techniques, and find the dimensions of box of maximum volume. Show all work and state conclusion.

Explanation / Answer

h=2*w since the height is twice the width. l+w+h=120 so plug in h=2*w to get l+w+2*w=120. Solve for l: l=120-3*w. We let w=x and use the formula for volume, and plug in all our known values: V(x)=l*w*h=(120-3*x)*(x)*(2*x). A)V(x)=240x^2-6*x^3. B)V'(x)=480*x-18*x^2. Set this equal to 0: 480*x-18*x^2=0. Factor out x to yield: x*(480-18*x)=0. x=0, x=480/18=26.667. So the width=26.67, length=120-3*26.67=40,and height=2*26.67=53.33.

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