# Thread: finding the value of x

1. ## finding the value of x

An open rectangular tank of height h metres with a square base of side x metres is to be constructed so that it has a capacity of 500 cubic metres. Prove that the surface area of the four walls and the base will be

2000 + x2 (the 2 is squared)

x

square metres. Find the value of x for this expression to be a minimum.

i have no idea where to even start on this one.

2. Are you saying it's equal to.........

$\displaystyle 2000 + x^2$

3. that just how the question was written, and like i say i have no idea

4. Oh right I see now.

Volume of the tank = 500

Volume = $\displaystyle x^{2}h$

Thus

$\displaystyle h = \frac{500}{x^2}$

Surface area minus lid = $\displaystyle 4hx + x^2$

Substitute $\displaystyle h = \frac{500}{x^2}$ into surface area equation to get.

$\displaystyle 4\times\frac{500}{x^2}x + x^2 = S$

Where S is the surface area of the tank base and four walls.

This simplifies to be

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

Happy new year.

Regards,
Ross

5. would it not be 5x instead of 4x on this bit

as there are 4 sides and a base so altogether 5 surfaces.?

6. No, because it's 4 walls are the same surface area but not the base.

Thus the base is still the $\displaystyle x^2$ and the four walls are represented by the other component.

7. oh right. cheers mate.