In this question, you will work step-by-step through an optimization problem. Many airlines restrict the size of carry on item in terms of their combined 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 50 inches". Suppose I want to take a box (cuboid) on my trip that will allow me to carry on as much as possible, and that I also want this box to have a square base. The questions we will answer using optimization are: What dimensions should the box have, and what would be the resulting maximum volume? If the base of the box is a square whose sides have length x, and the height of the box is h (both measured in inches), enter expressions for the volume of the box, V, and the sum of the length, width and height of the box in terms of x and h. V :  Sum L+W+H :    Since the sum of the length, width and height must be less than or equal to 50 inches, and it is clearly best to choose this sum to equal 50 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 =  What is the objective function for this problem? Using the constraint equation, rewrite the objective function in terms of x alone V =  Differentiate V with respect to x, to find the derivative dV / dx. dV/dx =  Find the values of x for which we have a potential relative extreme point of V. One of these is at x = 0, and the other occurs when x is equal to what value? (Give your answer, and those that follow, correct to two decimal places.) x =  For this question, you need not show that this is actually a relative maximum point. (But you should know how you would do this.) Now that you know the length of the base of the box, what is its height? height :  inches What is the maximum volume of the box? volume :  cubic inches