# Finding Derivitive with Expression of Area

1. Apr 9, 2012

### odolwa99

I have managed to solve (i), so I'll just post the answer as it comes into play for (ii). I'm struggling with the differentiation of this expression. Can anyone help?
Many thanks.

1. The problem statement, all variables and given/known data

Q. A tank, with a base, is made from a thin uniform metal. The tank, standing on level ground, is in the shape of an upright circular cylinder & hemispherical top, with radius length of r metres. The height of the cylinder is h metres. (i) If the total surface area of the tank is 45∏m2, express h in terms of r, (ii) Find the values of h & r, for which the tank has maximum volume.

2. Relevant equations

3. The attempt at a solution

Attempt: (i) $\frac{45 - 3r^2}{2r}$

(ii) 1st, separate the fractions and simplify the answer in (i) to $\frac{45}{2r}$ - $\frac{3r}{2}$
$\frac{dS}{dx}$ = -$\frac{45}{r^2}$ - $\frac{3}{2}$ = 0 => $\frac{3r^2}{2}$ = -45 => 3r2 = -90 => r2 = -30 => r = -$\sqrt{30}$

Ans: (From text book): r = 3

2. Apr 9, 2012

### scurty

Your formula you are trying to maximize is that of surface area, not volume!