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!

Question

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/cm2 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 = (5/3)(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.

Answer 1

If the pyramid has base side 2x and height h, then you know that it must have an area of 100 cm^2. The slant height of each face is √(h^2+x^2) so that means that
(2x)^2 + 4(x√(h^2+x^2)) = 100
h^2 = 25(25-2x^2)/x^2
Now, the volume is
v = 1/3 (2x)^2 h = 4/3 x^2 h = 20/3 x√(25-2x^2)
dv/dx = 20/3 (25-4x^2)/√(25-2x^2)
so max volume is when x = 5/2
That is, when the pyramid has a base with side = 5cm

Note that my solution is a bit different, since I used 2x for the base side to avoid fractions. But the answer is the same. You can massage it as you see fit.

Answer 2

Did you make a reasonable 3D sketch of the pyramid?

I let the base of the pyramid be 2x by 2x, (avoiding some fractions later on)
and let the height of the pyramid be y
So we know the volume = (1/3)(4x^2)(y) , let that one sit for a while

Look at one of the triangles, let its height be h,
then h^2 = x^2 + y^2 , (this is why I let the base be 2x)
h = √(x^2 + y^2)

I will assume the cost is $10/cm^2 or is it 10 cents/cm^2 , this will be a problem
I will assume it is $10

cost of 1 pyramid = 10(4x^2) + (10)(4)(1/2)(2x)(h) = 1000
4x^2 + 4xh = 100
x^2 + xh = 25
h = (25-x^2)/x

Now back to our h = √(x^2 + y^2)
√(x^2 + y^2) = (25-x^2)/x
square both sides:
x^2 + y^2 = (625 - 50x^2 + x^4)/x^2
x^4 + x^2 y^2 = 625 - 50x^2 + x^4
x^2 y^2 = 625 - 50x^2
y = √(625-50x^2)/x

and now back into V = (1/3)(4x^2)(y)
= (1/3)(4x^2)√(625 - 50x^2)/x
= (4/3) x √(625 - 50x^2)
= (4/3)√(625x^2 - 50x^4)

dV/dx = (4/3)(625x^2 - 50x^4)^(-1/2) (1250x - 200x^3) = 0
This can only be true if 1250x - 200x^3 = 0
50x(25 - 4x^2) = 0
x = 0 , which clearly would produce a minimum volume of 0
or
4x^2 = 25
x = 5/2

so the base is 5 by 5
y = √(625-50x^2)/x
= √(625 - 312.5)/2.5 = 7.071

I did not check my calculations, I should have written the solution out first
You better check this mess