I know that this problem is an optimization problem, and have been going through that section in my textbook. I feel like I understand the examples but am unsure how to begin with this problem or if I am on the right track with my calculations. Any direction or hints are appreciated!
Problem: You work for a fancy packaging company. Your client wants you to produce containers that will hold a very valuable mineral in powder form. The containers will be made of a material that costs $10$/cm$^2$ and you can spend $\$1000$ per container. Furthermore, the container needs to be in the shape of a pyramid with square base when folded up. Your task is to design a container (find the length of the base and height of the pyramid) that has the maximum possible volume, and costs exactly $\$1000$ to produce.
a. What facts/formulas will you need? (I said we would need to know how to find the first and second derivative, the formula for the surface area of a square and four triangles, the formula for the volume of a pyramid, and the first derivative test for Absolute Extreme Values as well as how to derive Extreme Values)
b. Write the equation that you need to optimize, and also an equation that describes the constraint. (I'm not sure how to write the equation when we use different formulas for area and volume, but I was thinking that it would be a problem where I set the surface area equal to the volume, and the constraint would be the cost has to be equal to $1000)
C. Show that the equation you need to maximize, as a function of a single variable x (the length of the pyramid's base) is $V = \frac53\sqrt{100x^2-2x^4}$
d. Maximize your equation either by using derivatives or by graphing or both. Then write down the dimensions of the pyramid (radius and height) that will lead to the pyramid with maximum volume possible (given your constraint). Indicate also the (now optimal) volume of your pyramid.
Look at this part of the question:
You're being asked to find the optimal values of two variables. First, the length of the base. After peeking at part C, let's call that $x$ for consistency. Second, the height of the pyramid. You can choose your own variable name for this; I recommend $h$.
Now for part B:
You should write an equation for the volume as a function of $x$ and $h$. Again, peeking ahead to part C, we see that the textbook is using a capital $V$ for the volume. So you want an equation like "$V=2x^3-7h$", except of course that's not the right equation.
Write an equation for the cost, let's call it $C$, like "$C=24(x+h)$". Then your constraint is $C=1000$.
You'll then be able to tackle part C by using the constraint to solve for $h$ as a function of $x$, and plugging that back into $V$. (This is not the only way to solve the problem, but it's the path you're being led down.)