A rectangular tank with a bottom and sided but but no top is to have volume 500 cubic feet. Determine the

dimensions (length, width, height) with the smallest possible surface area.

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- Mar 9th 2013, 12:06 PMapatitemax/min
**A rectangular tank with a bottom and sided but but no top is to have volume 500 cubic feet. Determine the**

dimensions (length, width, height) with the smallest possible surface area. - Mar 9th 2013, 02:13 PMILikeSerenaRe: max/min
Hi apatite! :)

For a minimal surface area, the top of the tank will be square.

Let's say the side of this square is x.

And let h be the height of the tank.

Then the volume is $\displaystyle x \times x \times h = 500$ and the surface area is $\displaystyle A = x \times x + 4 x \times h$.

Can you combine these equations to find A as a function of x?

Afterwards its derivative needs to be zero, which will give you x. - Mar 12th 2013, 06:54 AMapatiteRe: max/min
- Mar 12th 2013, 07:05 AMILikeSerenaRe: max/min
- Mar 12th 2013, 07:17 AMapatiteRe: max/min
- Mar 12th 2013, 07:19 AMILikeSerenaRe: max/min
- Mar 12th 2013, 07:24 AMapatiteRe: max/min
- Mar 12th 2013, 07:28 AMILikeSerenaRe: max/min
Looks like you did not substitute correctly.

$\displaystyle x \times x \times h = 500$

$\displaystyle h = \frac {500}{x \times x} \qquad (1)$

Substitute (1) into the formula for the area:

$\displaystyle A = x \times x + 4 x \times h$

$\displaystyle A = x \times x + 4 x \times \left(\frac {500}{x \times x}\right)$ - Mar 12th 2013, 07:46 AMapatiteRe: max/min
- Mar 12th 2013, 08:00 AMILikeSerenaRe: max/min
Not quite.

Multiplication has a higher priority than addition.

$\displaystyle A = x \times x + 4 x \times \left(\frac {500}{x \times x}\right)$

$\displaystyle A = x \times x + \left(\frac {4 x \times 500}{x \times x}\right)$

$\displaystyle A = x^2 + \frac {2000}{x}$