XP Math Games Arcade

11-26-2006

I have a hard time with an optimization problem. Any help is much appreciated.
Problem: A Yuppie from Yorkville wishes to build a whirlpool in the renovated loft. The whirlpool will be cylindrical in shape and will hold 5m cube of water. The bottom of the pool will be ceramic tile and the sides will be redwood. Ceramic costs twice as much as the wood. Find the dimensions of the pool to minimize the cost of construction.

Mr. Hui · 11-27-2006

Start with the formula for the surface area and volume of a cylinder.

$SA = 2{\pi}rh+2{\pi}r^2$

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

It holds 5m³ of water, so

${\pi}r^2h = 5$

and

$h = \frac{5}{{\pi}r^2}$

$SA = 2{\pi}r({\frac{5}{{\pi}r^2}})+2{\pi}r^2$

$S(r) = SA = {\frac{10}{r}}+2{\pi}r^2$

$S'(r) = SA' = -{\frac{10}{r^2}}+4{\pi}r$

${\frac{10}{r^2}} = 4{\pi}r$

$4{\pi}{r^3} = 10$

$r = \frac{\sqrt[3]{2.5}}{\sqrt[3]{\pi}}$

Use the $r$ to find the $h$ resulting in a cylinder with the smallest possible surface area.

**Sillysidley** · 11-27-2006

What grade level is this supposed to be? I could probably do it ( in like an hour...), but i will probably get it wrong

Mr. Hui · 11-27-2006

Calculus is most often a (required) college level mathematics course. It is also offered in high schools as Advanced Placement.

This particular example deals with 3-D optimization, a topic usually covered in an advanced calculus course.

11-28-2006

that stuff looks and i know it is, HARD.
WUT GRADE IS LILI2000 IN???
is she insane doing this kind of work

**Sillysidley** · 11-28-2006

Quote:

Originally Posted by kickonlanky

that stuff looks and i know it is, HARD.
WUT GRADE IS LILI2000 IN???
is she insane doing this kind of work

U will probably have to do this kind of stuff too later

**Sillysidley** · 11-28-2006

It sure is

**Sillysidley** · 11-28-2006

trig isn't too hard...

Mr. Hui · 11-28-2006

Trig is hard. It's mostly memorization of the sin, cos and tan functions.

**Sillysidley** · 11-28-2006

what is the correct name of that circle with a ine through it? It can be used in the equation s=r(the circle thin)

it is the central angle, but waht is the correct name?

11-26-2006	#1
Lili2000 Guest Posts: n/a	optimization problem calculus I have a hard time with an optimization problem. Any help is much appreciated. Problem: A Yuppie from Yorkville wishes to build a whirlpool in the renovated loft. The whirlpool will be cylindrical in shape and will hold 5m cube of water. The bottom of the pool will be ceramic tile and the sides will be redwood. Ceramic costs twice as much as the wood. Find the dimensions of the pool to minimize the cost of construction.

11-27-2006	#2
Mr. Hui Join Date: Mar 2005 Posts: 10,609	Start with the formula for the surface area and volume of a cylinder. $SA = 2{\pi}rh+2{\pi}r^2$ $V = {\pi}r^2h$ It holds 5m³ of water, so ${\pi}r^2h = 5$ and $h = \frac{5}{{\pi}r^2}$ $SA = 2{\pi}r({\frac{5}{{\pi}r^2}})+2{\pi}r^2$ $S(r) = SA = {\frac{10}{r}}+2{\pi}r^2$ $S'(r) = SA' = -{\frac{10}{r^2}}+4{\pi}r$ ${\frac{10}{r^2}} = 4{\pi}r$ $4{\pi}{r^3} = 10$ $r = \frac{\sqrt[3]{2.5}}{\sqrt[3]{\pi}}$ Use the $r$ to find the $h$ resulting in a cylinder with the smallest possible surface area. __________________ Do Math and you can do Anything! Last edited by Mr. Hui; 11-27-2006 at 09:14 PM..

11-27-2006	#3
Sillysidley Join Date: Oct 2006 Posts: 822	??????????? What grade level is this supposed to be? I could probably do it ( in like an hour...), but i will probably get it wrong __________________ .

11-27-2006	#4
Mr. Hui Join Date: Mar 2005 Posts: 10,609	Calculus is most often a (required) college level mathematics course. It is also offered in high schools as Advanced Placement. This particular example deals with 3-D optimization, a topic usually covered in an advanced calculus course. __________________ Do Math and you can do Anything!

11-28-2006	#7
Sillysidley Join Date: Oct 2006 Posts: 822	It sure is __________________ .