Question

Find the maximum volume of a right circular cylinder if the sum of the radius and the height of the given cylinder is 6?

• A. 28 p
• B. 32 p
• C. 64 p
• D. 40 p

Answer 1

The correct option is B

32 p

Conventional method

Volume of a cylinder, V = $\pi$r${}^2$h ( r = radius and h = height)

Given that r+h=6. Hence, h = 6-r

Hence, V = $\pi$r${}^2$(6-r)

V = 6$\pi$r${}^2$ - $\pi$r${}^3$

For maximizing volume, we need to differentiate V with respect to r and equate it to 0.

$\frac{dv}{dr}$ = 0

$\frac{dv}{dr}$ = 12pr - 3$\pi$r${}^2$ = 0

Hence, r = 4

R + h is given as 6, hence h = 2

V = $\pi$r${}^2$h = $\pi$$\times$4${}^2$$\times$2 = 32$\pi$

Alternate Method:

Volume of a cylinder = $\pi$r${}^2$h ( r =radius and h= height)

Given that r+h=6

To maximize $\pi$r${}^2$h, we need to maximize r${}^2$h which happens when $\frac{r}{2}$ = $\frac{h}{1}$,

$\Rightarrow$ r = 4 and h =2

Maximum volume = $\pi$$\times$4${}^2$$\times$2 = 32$\pi$

Points to remember

1. If a+b=constant, ab will be maximum when a=b

2. If ab=constant, the minimum value of a+b will be obtained at a=b

3. If a+b+c is a constant, then a${}^m$. b${}^n$ .c${}^p$ is maximum when $\frac{a}{m}$ = $\frac{b}{n}$ = $\frac{c}{p}$ (the above question is an example of this)