Question

Assertion :Water is dropped at the rate ${2m}^3/sec$ into a cone of semivertical angle of ${45}^0$.

The rate at which periphery of water surface changes when height of water in the cone is $2$ meters is $1m/sec.$ $r=2$.

Reason: The rate of change of radius of water surface is $\frac{1}{2\pi }$

• 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 incorrect, Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given that $\frac{dv}{dt}=2$
$\frac{d}{dt}\left(\frac{{\pi r}^3}{3}\right)=2\Rightarrow \pi r^2\frac{dr}{dt}=2$
$\Rightarrow \frac{dr}{dt}=\frac{2}{{\pi r}^2}\Rightarrow \frac{d}{dt}\left(2\pi r\right)=\frac{4}{r^2}$
when $h=2m\Rightarrow r=2m$
hence $\frac{dp}{dt}=\frac{4}{4}=1m/sec$.
Thus the assertion and reason are correct and the reason is the correct explanation for the assertion.