Question

A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(2x+3)$. The rate of change of volume with respect to $x$ will be

• A. $\frac{27\pi }{8}(2x-3{)}^2$
• B. $\frac{27\pi }{8}(2x+3{)}^2$
• C. $\frac{27\pi }{8}(3x-2{)}^2$
• D. $\frac{8}{27\pi }(2x+3{)}^2$

Answer 1

The correct option is B $\frac{27\pi }{8}(2x+3{)}^2$

Given diameter $2r=\frac{3}{2}(2x+3)$
$\Rightarrow r=\frac{3}{4}(2x+3)$
$\Rightarrow \frac{dr}{dx}=\frac{3}{4}\times 2=\frac{3}{2}$
$V=\frac{4}{3}\pi r^3$
$\Rightarrow \frac{dv}{dx}=\frac{4}{3}\pi \times 3r^2\times \frac{dr}{dx}$
$\Rightarrow \frac{dV}{dx}=4\pi \times \frac{9}{16}(2x+3{)}^2\times \frac{3}{2}$
$\frac{dV}{dx}=\frac{27\pi }{8}(2x+3{)}^2$