Answer to Question #135999 in Calculus for Sean

Question

A closed cylindrical tank with radius r and height h has a surface area of 100 sq. units.
Express its volume in terms of r.

Answer 1

We have given,

Since, surface area of the cylinder is sum of the surface area of curved surface and two circular region (One is up called base and another is top ).

Thus,

$$A=2\pi r^2+2\pi rh\\ \implies A= 2\pi r(r+h)$$

In this case we have $A=100$ , thus

$$100=2\pi r^2+2\pi rh\implies h=\frac{50-\pi r^2}{\pi r}$$

Now, we know that Volume of Cylinder is

$$V=\pi r^2 h\\ \implies V=\pi r^2\bigg(\frac{50-\pi r^2}{\pi r}\bigg)=r(50-\pi r^2)$$