Question

The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is $1cm$ the altitude is $6cm$. When the radius is $6cm$, then volume is increasing at the rate of $1$ Cu $cm/sec$. When the radius is $36cm$, the volume is increasing at a rate of $n$ cu. $cm/sec$. The value of '$n$' is equal to:

• A. $12$
• B. $22$
• C. $30$
• D. $33$

Answer 1

Answer: (D)

$33$

Let $r$ is the radius ,$h$ is altitude and $v$ is the volume of right-circular cylinder
$\because \frac{dr}{dt}=c$ and $h=ar+b$ where $a,b,c$ are a constant
$\because h=ar+b$ and $\frac{dh}{dt}=3\frac{dr}{dt}\Rightarrow \frac{dh}{dt}=3c$
Using Chain Rule $\frac{dh}{dr}.\frac{dr}{dt}=3c\Rightarrow ac=3c$
$\Rightarrow a=3$ assuming $c\ne 0$
$\Rightarrow h=3r+b$
when $r=1,h=6\Rightarrow 6=3\times 1+b\Rightarrow b=3\Rightarrow h=3r+3$
As $v=\pi r^{2}h$ for a right circular cylinder
$\Rightarrow \frac{dv}{dt}=\pi \left[\right(2r)\frac{dr}{dt}h+r^2\frac{dh}{dt}]$
When $r=6,\frac{dv}{dt}=1\Rightarrow c=8.84\times {10}^{-4}$
When $r=36,\frac{dv}{dt}=n$ and putting $c=8.84\times {10}^{-4}$
Thus, $\frac{dv}{dt}=n=\pi (7992\times c+3888\times c)$
$\Rightarrow n=33$