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Mathematics · 2026 · 3 marks
CBSE 2026 · Region 5 · Set 1 · Q26
A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface…
A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.
Marking-scheme solution
Let r be the radius, $\displaystyle V=\dfrac{4}{3} \pi r^{3}$ be the volume
$\displaystyle \dfrac{dV}{dt}=-k\left(4 \pi r^{2}\right)$ where $\displaystyle k>0$
$\displaystyle 4 \pi r^{2} \dfrac{dr}{dt}=-4 k \pi r^{2}$
$\displaystyle \therefore \dfrac{dr}{dt}=-k$ which is a constant
Thus, radius is decreasing at a constant rate
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