Question

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

• A. $11$
• B. $22$
• C. $33$
• D. $44$

Answer 1

The correct option is C $33$

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, $\frac{dv}{dt}=n=\pi (7992\times c+3888\times c)$

$\Rightarrow n=33$