Question

Diagram 5 shows a right cylinder with a diameter of 2xcm. Given that the total surface area of the cylinder is 96cm³.Find the maximum volume of the cylinder

Answer 1

Given:

The diameter of the right cylinder is 2x cm.

The total surface area is 96 cm cube.

The radius is calculated as,

`\begin{gathered} r=(d)/(2) \\ r=(2x)/(2) \\ r=x\text{ cm} \end{gathered}`

The total surface area is,

`\begin{gathered} S=2\pi rh+2\pi(r)^2 \\ 96=2\pi xh+2\pi(x^2) \\ h=(96-2\pi(x^2))/(2\pi x) \end{gathered}`

Volume is,

`\begin{gathered} V=\pi(r)^2h \\ =\pi(x^2)(96-2\pi(x^2))/(2\pi x) \\ =(x(96-2\pi(x^2))/(2) \end{gathered}`

Now, differentiate with respect to x,

`\begin{gathered} (dV)/(dx)^{}=(d)/(dx)((x(96-2\pi(x^2))/(2)) \\ =(d)/(dx)\mleft(x\mleft(-\pi x^2+48\mright)\mright) \\ =(d)/(dx)\mleft(x\mright)\mleft(-\pi x^2+48\mright)+(d)/(dx)\mleft(-\pi x^2+48\mright)x \\ =1\cdot\mleft(-\pi x^2+48\mright)+\mleft(-2\pi x\mright)x \\ =84-3\pi(x^2)\ldots\ldots\ldots\ldots\text{.}(1) \end{gathered}`

Now,

`\begin{gathered} (dV)/(dx)=0 \\ 84-3\pi(x^2)=0 \\ x^2=(16)/(\pi) \\ x=\sqrt[]{(16)/(\pi)} \end{gathered}`

Now, differentiate (1) with respect to x again,

`\begin{gathered} (d^2V)/(dx^2)=(d)/(dx)(84-3\pi(x^2)) \\ =-6\pi x \\ At\text{ x=}\sqrt[]{(16)/(\pi)} \\ (d^2V)/(dx^2)=-6\pi\sqrt[]{(16)/(\pi)}<0 \\ \end{gathered}`

Since, the double derivative is negative.

`So,\text{ the volume is maximum at }\sqrt[]{(16)/(\pi)}`

So, the volume becomes,

`\begin{gathered} V=\pi(x^2)h \\ V=\pi(\sqrt[]{(16)/(\pi)})^2h \\ V=(16h)/(\pi) \end{gathered}`

Answer: maximum volume of the cylinder is,