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Question

An open box is to be made out of a 6-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Answer 1

This is a classic optimization question. In optimization problems, we follow some easy steps:

1. Draw it out! If the problem is geometric in nature, visualizing it can help a lot
2. Write out a formula for the thing you're trying to optimize
3. Take the derivative of that formula and set it equal to 0
4. Find the solutions and plug them back into the original formula to see which one gives the optimal answer (biggest volume, fastest time, etc.)

So for this problem, the steps would look like this:

1. Draw out the rectangle with squares cut out of each corner, and label what you know
2. Since we are cutting out a square from each corner, that means the long side is 14-2x (there are 2 pieces of x length cut from that side) and the short side is 6-2x
3. Now draw a rectangular prism representing folding the cardboard to make an open box
4. It should have a length of 14-2x, a width of 6-2x, and a height of x
5. The thing we're trying to optimize is the volume of a box, so our formula is V=LWH (length * width * height), but we want just 1 variable on the right side, so we write it as V=(14-2x)(6-2x)(x) since we know those are the length, width, and height
6. To take the derivative, we could use the product rule, but it's easier to just simplify the volume function first by FOILing: V=[(14)(6)+(14)(-2x)+(-2x)(6)+(-2x)(-2x)]x and further simplifying V=84x-40x²+4x³; now we take the derivative which is simply the power rule to get dV/dx=84-80x+12x²
7. We set that equal to 0 and solve for x, which you can do by factoring, using the quadratic formula, or graphing and finding the x-intercepts of the curve

Since this is a parabola, we'll likely get 2 solutions, so we plug both of those back in for x in our ORIGINAL VOLUME EQUATION (not the derivative!) to see which of those x-values gives us the largest volume

HINT: If you think about the original dimensions of the cardboard, one of the x-values might not make any sense (remember, we're cutting off TWO lengths equal to x from each side, and the shortest side of the cardboard is only 6 inches long)

Answer 2

V(x) = x(14 - 2x)(6 - 2x)

Expand, take first derivative, set eqaul to zero, solve for x.