A balloon remains spherical with diameter $\dfrac{3}{2}(2x+1)$. The ra…

Question

A balloon remains spherical with diameter $\dfrac{3}{2}(2x+1)$. The rate of change of its volume with respect to $x$ is

Accepted Answer

A. $\dfrac{27\pi}{8}(2x+1)^2$

1. Radius $=\dfrac{1}{2}\cdot\dfrac{3}{2}(2x+1)=\dfrac{3}{4}(2x+1)$.
2. $V=\dfrac{4}{3}\pi r^3=\dfrac{4}{3}\pi\cdot\dfrac{27}{64}(2x+1)^3=\dfrac{9\pi}{16}(2x+1)^3$.
3. $\dfrac{dV}{dx}=\dfrac{9\pi}{16}\cdot 3(2x+1)^2\cdot 2=\dfrac{27\pi}{8}(2x+1)^2$.
4. The trap $(2x+1)^3$ forgets to differentiate.
_Source: NCERT Class 12 Mathematics Ch 6 "Application of Derivatives", p.4_

Answer

$\dfrac{9\pi}{8}(2x+1)^2$

Answer

$\dfrac{27\pi}{16}(2x+1)^2$

Answer

$\dfrac{27\pi}{4}(2x+1)^3$