# Can someone help me with a calculus word problem?

• May 5th 2009, 07:19 AM
anitah
Can someone help me with a calculus word problem?
I really don't know how to do the following problem:

A conical bin must have a volume of 2500 ft3. The circular base is made of a concrete slab, which will cost $1.30 per square foot, and the material for the side costs$3.50 per square foot. Find the dimensions that minimize the total cost.

I am in an introductory math course and he said to solve this using calculus... how do I start? Am I supposed to take the floor cost multiplied by the side cost? If anyone could give me some helpful hints or the correct answer, I'd really appreciate it. Thanks.
• May 5th 2009, 10:34 AM
derfleurer
$\displaystyle V = \pi r^2h = 2500$

You want a function for cost.

$\displaystyle V' = SA = 2\pi rh + \pi r^2$

Where the cost of the side is 3.5 and the ends .65 (to account for only one base at 1.30)

$\displaystyle C = 2(3.5)\pi rh + 0.65\pi r^2$

$\displaystyle h = \frac{V}{\pi r^2}$

$\displaystyle C = 2(3.5)\pi r\frac{V}{\pi r^2} + 0.65\pi r^2 = \frac{7V}{r} + 0.65\pi r^2$

$\displaystyle C' = \frac{-7V}{r^2} + 1.30\pi r$

$\displaystyle C' = \frac{1.30\pi r^3 - 7V}{r^2}$

$\displaystyle \frac{1.30\pi r^3 - 7V}{r^2} = 0$

$\displaystyle 1.30\pi r^3 - 7V = 0$

Plug in for V, finish calculating r, get your dimensions.
• May 5th 2009, 10:52 AM
skeeter
hate to spoil such a great response, but ...

Quote:

A conical bin must have a volume ...
• May 5th 2009, 11:56 AM
derfleurer
Wow. =p Well, I'm pretty sure my 0.65 should've stayed a 1.3 for the cost function.

I'll make another response soon if someone else doesn't decide to.