Question

If a spherical balloon has a variable diameter $3x+\frac{9}{2},$ then the rate of change of its volume with respect to $x$ is

• A. $27\pi (2x+3{)}^2$
• B. $\frac{27\pi }{16}(2x+3{)}^2$
• C. $\frac{27\pi }{8}(2x+3{)}^2$
• D. None of these

Answer 1

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

$d\left(x\right)=3x+\frac{9}{2}$
$\Rightarrow r\left(x\right)=(\frac{3x}{2}+\frac{9}{4})$
$\therefore V\left(x\right)=\frac{4}{3}\pi {(\frac{3x}{2}+\frac{9}{4})}^3$
$\frac{dV}{dx}=\frac{4\pi }{3}\times 3{(\frac{3x}{2}+\frac{9}{4})}^2\times \left(\frac{3}{2}\right)$
$\Rightarrow \frac{dV}{dx}=6\pi \times \left(9\right)\times \frac{(2x+3{)}^2}{16}$
$\Rightarrow \frac{dV}{dx}=\frac{27}{8}\pi (2x+3{)}^2$