1. ## 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).

2. Find the minimum distance from the parabola
to the point (0,-3).
$y=\pm\sqrt{x}$

$L=\sqrt{(x-0)^{2}+(y+3)^{2}}$

$L=\sqrt{x^{2}+(-\sqrt{x}+3)^{2}}$

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

$S=L^{2}=x^{2}+(-\sqrt{x}+3)^{2}$

$\frac{dS}{dx}=2x-\frac{3}{\sqrt{x}}+1$

$2x-\frac{3}{\sqrt{x}}+1=0$

Now, solve for x.

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

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

4. 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?

5. 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:

$V=x(10-2x)(14-2x)$

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?.