The radius of spherical balloon is decreasing at the rate of 0.1 cm/sec, the rate at which its volume is decreasing, when its radius is 0.5 cm is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\pi}{10}\,cm^3/sec\)
The volume of a sphere of radius r is
\(\displaystyle V = \frac{4}{3}\pi r^3\)
Differentiating with respect to time t by the chain rule:
\(\displaystyle \frac{dV}{dt} = 4\pi r^2\,\frac{dr}{dt}\)
The radius is decreasing, so \(\frac{dr}{dt} = -0.1\) cm/sec. Putting r = 0.5:
\(\displaystyle \frac{dV}{dt} = 4\pi(0.5)^2(-0.1)\)
\(= 4\pi(0.25)(-0.1) = -0.1\pi\)
\(\displaystyle = -\frac{\pi}{10}\)
The negative sign shows the volume is decreasing, at the rate of \(\frac{\pi}{10}\) cm3/sec.