Thread: Optimization

A box is to be made out of a 10 by 14 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W is less than or requal to L).


Find two numbers A and B(with A is less than or equal to B) whose difference is 32 and whose product is minimized.


Find the minimum distance from the parabola

to the point (0,-3).

Find the minimum distance from the parabola

to the point (0,-3).

There is a trick that is helpful in minimizing or maximizing a distance. It is

based on the observation that the distance and the square of the distance

have their max or min at the same point. Therefore, we can do it sans radical







Now, solve for x.

If you want to determine the nature of this critical point, use the second derivative test.

I got 1 for x, but it isn't right. Any ideas?

I figured it out, y=-1 and x=1, so 2.2361 is the answer out of the equation. Any help on the other two?

A box is to be made out of a 10 by 14 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W is less than or equal to L).

These problems are all very cliche. If you google them, you will probably find them or, at least, something very similar.

Let x be the length and width of the square cut out of each corner.

Then, when we fold them up the volume would be:



When you differentiate, set to 0 and solve for x, you will get two solutions. Only one will be in the domain of the problem.

#2 is very straighforward. Try that one yourself. OK?.