Question

Assertion :Water dropped at the rate of $2m^3/sec$ in to a cone of semi-vertical angle of ${45}^0$. The rate at which periphery of the water surface changes with height of water in the cone of $2m$ is $2m/sec.$ Reason: The rate of change of periphery of water surface is $\frac{d}{dt}\left(2\pi r\right)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is false but Reason is correct

Answer 1

The correct option is D Assertion is false but Reason is correct

Given rate of change of volume i.e. $\frac{dv}{dt}=2$
m${}^3$/sec volume of cone$,v=\frac{1}{3}\pi r^{2}h$
$\because \theta ={45}^0=>h=r$
$\therefore$ $v=\frac{1}{3}\pi r^3$
$\Rightarrow$ $\frac{dv}{dt}=\frac{d}{dt}\left(\frac{1}{3}\pi r^3\right)=2$
$\Rightarrow$ $\pi r^2\frac{dr}{dt}=2$, $\frac{dr}{dt}=\frac{2}{\pi r^2}$
Now $\frac{dP}{dt}=\frac{d}{dt}\left(2\pi r\right)=2\pi \frac{dr}{dt}=\frac{2\pi \times 2}{\pi r^2}=\frac{4}{r^2}=1$, (as $h=2$ m then $r=2$)
$\therefore$ Assertion is false.