Question

A sphere of radius 100 mm shrinks to radius 98 mm, then the approximate decrease in its volume is
(a) 12000 π mm³
(b) 800 π mm³
(c) 80000 π mm³
(d) 120 π mm³

Answer 1

(c) 80000 π mm³

Let x be the radius of the sphere and y be its volume.

$$\begin{gathered} x=100,x+∆x=98 \\ \Rightarrow ∆x=-2 \\ y=\frac{4}{3}\pi x^3 \\ \Rightarrow \frac{dy}{dx}=4\pi x^2 \\ \Rightarrow {\left(\frac{dy}{dx}\right)}_{x=100}=40000\pi \\ \therefore ∆y=dy=\frac{dy}{dx}dx=40000\pi \times \left(-2\right)=-80000\mathrm{\pi } \\ \mathrm{Hence},\mathrm{the}\mathrm{decrease}\mathrm{in}\mathrm{the}\mathrm{volume}\mathrm{of}\mathrm{the}\mathrm{sphere}\mathrm{is}80000\mathrm{\pi }{\mathrm{mm}}^3. \end{gathered}$$