Question

I need help with this project. see http://imgur.com/DCKOP
I specifically need help with the first two parts, the last two I can figure out on my own for sure.

I've already figured out the restraint equation

girth + longest dimension <= 180 inches
height - diameter > 0

But I'm not sure what to do for payoff function or other parts.

Answer 1

Rectangular prism with square base of sidelength n:
4n +h = 180
h = 180-4n
Volume = n^2*h = n^2(180-4n) = 180n^2 -4n^3
To optimize this, take the derivative and solve for 0. This gives you your critical points. I think there'll be two of them. Check the volume at the critical points.

Cylinder with radius R:
2piR + h = 180
h = 180-2piR
Volume = piR^2*h = piR^2(180-2piR) = 180piR^2 - 2pi^2R^3
Take the derivative and solve for 0 to get the critical points. Check the volume at the critical points.

Answer 2

This may seem like a stupid question, but what is a derivative?

Answer 3

Hmmm... have you taken any calculus courses?

Answer 4

Nope! Just Trig and Pre Calc.

Answer 5

It's strange that you're being asked this question in a course that doesn't have calculus as a prerequisite or corequisite...

Answer 6

That's just how things work out I guess.
I'll figure it out, can't be too complicated.

Answer 7

Okay well let me give you the short version. A derivative tells you how fast a function is increasing at any point. So, if you know f(x) and you know the derivative, written as f'(x), then f'(x) tells you whether the function is increasing or decreasing and how quickly.

Here's a quick example:
For the function f(x) = 3x^2 - 12x
f'(x) = 6x -12
So if I want to know who quickly the function is increasing at x=1, I plug in
f'(1) = 6(1)-12 = -6 and that tells me f(x) is going down at that point.
If I check f'(2), I get
f'(2) = 12-12 = 0, which says that f(x) isn't changing either way at that point.
If I check f'(3), I get
f'(3) = 18-12 = 6, which says that f(x) is increasing at that point.

Here's what the function looks like