Question

A right circular cylinder which is open at the top and has a given surface area, will have the greatest volume if its height h and radius r are related by

• A. 2h = r
• B. h = 4r
• C. h = 2r
• D. h = r

Answer 1

Answer: (D)

h = r

Answer 2

Volume of cylinder, $(V) = \pi r^2h;$ Surface area, $(S) = 2 \pi rh + \pi r^2$ ......(1) $\Rightarrow h = \frac{S - \pi r^{2}}{2 \pi r}$ $\therefore V = \pi r^{2} \left[\frac{S - \pi r^{2}}{2 \pi r}\right] = \frac{r}{2} \left[S - \pi r^{2}\right] = \frac{1}{2} \left[Sr - \pi r^{3}\right]$ Now, Differentiate both sides, w.r.t 'r' $\frac{dV}{dr}= \frac{1}{2} \left[S - 3\pi r^{2}\right]$ Now, circular cylinder will have the greatest volume , when $\frac{dV}{dr} = 0$ $\Rightarrow S = 3\pi r^{2}$ $\Rightarrow 2\pi rh + \pi r^{2} = 3\pi r^{2} \Rightarrow 2\pi rh = 2\pi r^{2} \Rightarrow r =h$