Maximization problem

Question

A box with an open top is to be constructed from rectangular sheet of cardboard of size 5 feet by 8 feet. You are to cut out 4 corner squares of the cardboard in such a way that the volume of the box is as large as poossible. Let x denote the length in feet of the side of the square cutout.

A) write the volume of the box as a function of x.

b) what is the domain of the function in part a?

c) what is the derivative of the function in part a?

d) for what values of x is the derivative equal to 0?

e) what does your answer in part d tell you about the slope of the tangent line to the actual graph of the function of part a?

Answer 1

This is a start

V(x) = (5 - 2x)(8 - 2x)(x)

Can you provide answers to the remaining portion?

Answer 2

You start with a piece of cardboard 5 feet by 8 feet. You cut out from each corner of the original piece a square of dimension x foot by x foot. Since you cut a corner from each end, the cardboard that is left is, respectively, 5-2x feet and 8-2x feet. The box will be x feet high, so the total volume of the box will be

V = x(5-2x)(8-2x) cubic feet, so V = 4x^3-26x^2+40x cubic feet.

The domain of the function is determined by the fact that you are cutting squares from each corner of the original cardboard. In order to form a box at all, you need to cut some portion out of the corner (otherwise you couldn't fold the edges up), so x must be greater than zero. However, the length of the short side is 5 feet. Once the squares are cut from each corner, the remaining short side is 5-2x feet. Once again, this cannot equal zero (or you would not have an edge to bend up to form the box), so x must therefore be less than 2.5 feet. The domain is therefore 0 < x < 2.5.

The derivative of the volume function is 12x^2-52x+40

To find the values of x for which the derivative would equal to zero, set 12x^2-52x+40 =0. You can divide by 4 to simplify the equation to 3x^2-13x+10 =0. This can be factored to (3x-10)(x-1) =0. Setting each portion equal to zero gives 3x-10 =0 or x-1=0. Therefore, the values of x for which the derivative is zero are x=10/3 or x=1. Note that x=10/3 does not fall within the allowed domain and should be excluded. Therefore, x=1 is the only allowed value.

For a maximum or minimum problem, the answer corresponds to a value of x for which the tangent line is horizontal (slope = 0). A plot of the original volume function V=4x^3-26x^2+40x will show a horizontal tangent line at x=1.