Question

Many airlines restrict the size of carry on item in terms of theircombined length, width and height (L+W+H).
I am due to fly on an airline that states "The sum of the length,the width and height of your item must be at most 70 inches".
Suppose I want to take a box (cuboid) on my trip that will allow meto carry on as much as possible, and that I also want this box tohave a square base.
The questions we will answer using optimization are: Whatdimensions should the box have, and what would be the resultingmaximum volume

a.If the base of the box is a square whose sides have length x, andthe height of the box is h (both measured in inches), enterexpressions for the volume of the box, V, and the sum of thelength, width and height of the box in terms of x and h.
V :

Sum L+W+H :


b.Since the sum of the length, width and height must be less thanor equal to 70 inches, and it is clearly best to choose this sum toequal 70 inches, write down the constraint equation.
Rearrange the constraint equation to give the height of the box, h,in terms of the length of the base, x

h =

c.What is the objective function for this problem?
Using the constraint equation, rewrite the objective function interms of x alone

V =

d.Differentiate V with respect to x, to find the derivative dV /dx.
=

e.Find the values of x for which we have a potential relativeextreme point of V.

One of these is at x = 0, and the other occurs when x is equal towhat value?

(Give your answer, and those that follow, correct to two decimalplaces.)

x =

For this question, you need not show that this is actually arelative maximum point. (But you should know how you would dothis.)

f.Now that you know the length of the base of the box, what is itsheight?
height : inches

g.What is the maximum volume of the box?
volume : cubic inches