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 $6cm$, the volume is increasing at the rate of $1Cucm/sec$. When the radius is $36cm$, the volume is increasing at a rate of $ncu.cm/sec$. The value of ' $n$ ' is equal to:

• A. 12
• B. 22
• C. 30
• D. 33

Answer 1

Answer: (D)

$\frac{dr}{dt}=c\text{ and }h=ar+b;\text{ also }\frac{dh}{dt}=3\frac{dr}{dt}\text{ (given) }$
$\therefore a\frac{dr}{dt}=3\frac{dr}{dt}\Rightarrow a=3$
$\text{ hence }h=3r+b$
$\text{ when }r=1;h=6\Rightarrow 6=3+b\Rightarrow b=3$
$\therefore h=3(r+1)$
$V=\pi r^{2}h=3\pi r^2(r+1)=3\pi \left(r^3+r^2\right)$
$\frac{dV}{dt}=3\pi \left(3r^2+2r\right)\frac{dr}{dt}$

$\text{ where }r=6;\frac{dV}{dt}=1cc/\sec$
$\therefore 1=3\pi (108+12)\frac{dr}{dt}\Rightarrow 360\pi \frac{dr}{dt}=1$
$\text{ again when }r=36,\frac{dV}{dt}=n$
$n=3\pi \left((3.36{)}^2+2.36\right)\frac{dr}{dt}$
$n=3\pi \cdot 36(110)\cdot \frac{1}{360\pi }$
$n=33\Rightarrow \text{ (D) }$