Question

The radius of a sphere shrinks from $10$ to $9.8$ cm. The approximate decrease$($in cm${}^3)$in its volume is

• A. $20\pi$
• B. $40\pi$
• C. $60\pi$
• D. $80\pi$

Answer 1

The correct option is D $80\pi$

We have $x=10$ and $\Delta x=-0.2$

The volume of a sphere of radius $x$ is given by $V=\frac{4}{3}{\pi x}^3$

$\Rightarrow \frac{dV}{dx}=\frac{4\pi }{3}\left(3x^2\right)$

$\therefore dV=4\pi x^{2}dx$

$\Rightarrow dV=\left(400\pi \right)(-0.2)$

$\therefore dV=-80\pi$

Thus, the approximate decrease in the volume of the sphere is $80\pi {cm}^3$.