Question

I've been pulling my hair out with this one:

Optimization question:

- Creating a dorm, it must be 225,000 ft^(3)
- The dorm is in the shape of an airplane hangar, a half cylinder, so the volume formula is (pi*r^(2))/2
- Flooring is 30 per foot, sides are 20, roofing is 15
- the cost must be minimized

Now, the answer is:
(l/r)((27000000/pi)+6750000) + 20(pi)r^(2)
in which r = 49.611 and l = 58.1978

However, I don't know what steps to take to get there. Can anyone help?

Answer 1

The volume of the hanger is\[V=\frac\pi2r^2\ell=2.25\times10^5\]and the material needed will be\[A=\pi r\ell+2r\ell\]agreed so far?

Answer 2

yes

Answer 3

wait a minute, it says the roof cost different than the sides, how is that possible if it is a hanger
what is the side as opposed to the roof?

Answer 4

oh, we need the ends of the cylinder too, so I have to modify it I guess

Answer 5

yeah, the ends are the sides, i assume

Answer 6

\[V=\frac\pi2r^2\ell=2.25\times10^5\]\[A=\pi r\ell+2r\ell+\pi r^2\]no with the prices we can get a cost function from the area\[P=15\pi r\ell+60r\ell+20\pi r^2\]now do you agree ?
(really, don't just say 'yes' unless you do)

Answer 7

now with the prices*

Answer 8

With your area function, isn't the area function for a cylinder
\[2(\pi)r ^{2} + 2(\pi)rh\]

so for a hangar it's just half? where does the 2rl come from?

Answer 9

that is the area of the floor

Answer 10

ohhh, I see.

Answer 11

thanks for that! :P

Answer 12

welcome!
so now we need all this business in terms of one variable, which we can achieve by applying the restriction on the volume

Answer 13

\[V=\frac\pi2r^2\ell=2.25\times10^5\to \ell=\frac{4.5\times10^5}{\pi r^2}\](I could have solved it for r, but then we would have had a radical pi and I thought that looked ugly)
so now we can plug this expression for l into our cost function and it will be in terms of one variable...

Answer 14

that's going to be ugly, but I follow you.

Answer 15

actually, it's canceling out

Answer 16

yeah I'll do the algebra on paper and we can compare results; too hard to type

Answer 17

yay, I got the answer

Answer 18

thank you!!!!

Answer 19

oh good, I didn't want to hunt for a mistake in the mess that follows
congrads! happy to help :D