It's been overs ten years since my last calculus course, and I think I've regressed back to a pre-algebra level. Not looking for an answer here, just help getting started.
What I think I have here is a simultaneous equation, but I'm not sure... as I try to work it out, I keep getting stuck with what seems like a quadratic equation. I'm sure I'm wrong on one or both counts, so can someone lend some perspective?
I'm trying to design a cylinder with dimensions proportional to a larger cylinder.
The diameter of the original cylinder = 26.24 ft
The height of the original cylinder = 12.08 ft
This gives a diameter to height ratio of 2.172.
I'd like to find the dimensions necessary to design a 0.668 cubic foot cylinder while maintaining the proportions of the larger cylinder, so I've come up with the following simultaneous equations:
diameter = X (in feet)
height = Y (in feet)
(X/2)2 * PI * Y = 0.668 cubic feet
X/Y = 2.172
I've tried to solve for X in the first equation...
X2/4 * PI * Y = 0.668
Multiply LHS and RHS by 4 to clear demoninator
X2 * 4PI * 4Y = 2.672
Divide both sides by 4PI
X2 * 4Y = 2.672 / 4PI
X2 * 4Y = 0.2126
Divide both sides by 4Y
X2 = 0.05316 / Y
X = SQRT (0.05316 / Y)
... and then insert it into the second equation...
(SQRT (0.05316 / Y)) / Y = 2.172
Multiply RHS by Y to clear denominator
(SQRT (0.05316 / Y)) = 2.172Y
Square both sides
(SQRT (0.05316 / Y))2 = 4.7176Y2
0.05316 / Y = 4.7176Y2
Multiply both sides by Y
0.05316 = 4.7176Y3
Y3 = 0.0113 feet
Y = 0.224 feet
This would make the height of the cylinder 2.69 inches, and the diameter 5.84 inches. This isn't right.
I'm sure this is simple and that I'm making it overly complex. Still, any help would be really appreciated. Thanks!