Question

A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be $k$ times the height of the jar. The value of $k$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Let the diameter be D and the height be H.

The surface area S of the metal sheet, which is constant, can be expressed as the sum of the lateral surface area and the base of the cylinder.

$S=\pi DH+\pi \frac{D^2}{4}$

Hence.

$H=\frac{1}{\pi D}(S-\frac{\pi D^2}{4})$

$=\frac{S}{\pi D}-\frac{D}{4}$

The capacity, or volume, V of the cylinder is given as:

$V=\pi \frac{D^2}{4}H$

$=\pi \frac{D^2}{4}(\frac{S}{\pi D}-\frac{D}{4})$

$=\frac{DS}{4}-\frac{\pi D^3}{16})$

For the maxmium volume,

$\frac{dV}{dD}=0$

$\therefore \frac{S}{4}-\frac{3\pi D^2}{16}=0$

$S=\frac{3\pi D^2}{4}$

We showed earlier that

$H=\frac{S}{\pi D}-\frac{D}{4}$

$\therefore H=\frac{3D}{4}-\frac{D}{4}=\frac{D}{2}$

Hence, the diameter D is 2 times the height.

Answer: option B.