"Design a cylindrical tank with..." Volume&Surface Area help required.

**Baby Bat** · May 15th 2008, 04:53 PM

Question: You are in charge of a tank to be used for water storage. Your boss has told you that he wants a cylindrical tank with a minimum capacity of 300,000 cubic feet. Your boss wants to spend the least possible amount painting the tank. Design a tank to meet this objective.

I know I have to use volume and surface area to solve this, but I'm not sure how to go about it. Word problems and me just don't get along very well.

**Mathstud28** · May 15th 2008, 05:22 PM

Originally Posted by Baby Bat

Question: You are in charge of a tank to be used for water storage. Your boss has told you that he wants a cylindrical tank with a minimum capacity of 300,000 cubic feet. Your boss wants to spend the least possible amount painting the tank. Design a tank to meet this objective.

I know I have to use volume and surface area to solve this, but I'm not sure how to go about it. Word problems and me just don't get along very well.

You have your set equation Volume MUST be 300,000 cubic feet

$\therefore{300,000=\pi{r^2}h}$

And you want to minimize surface area...so let S denote surface area you have

$S=2\pi{r^2}+2\pi{r}h=2\pi{r}(r+h)$

So solve $300,00=\pi{r^2}{h}\Rightarrow{\frac{300,000}{\pi{r ^2}}=h}$

So now imputting that into our surface are equation we get

$S=2\pi{r}\bigg(r+\frac{300,000}{\pi{r^2}}\bigg)$

Now do you know calculus? If so Find $S'$ and find r such that $S'(r)=0$

After that verify that it is a minimum point by showing $S''(r)>0$

After you have verified this plug r back into your volume equation and solve for h

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