Question

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

• A. \(\frac{27π}{4}(2x+3)^2\)
• B. \(\frac{2π}{3}(2x+3)^2\)
• C. \(\frac{27π}{8}(2x+3)^2\)
• D. \(\frac{27π}{2}(2x+3)^2\)

Hint:

Write the radius from the diameter, find the volume in terms of x, then differentiate.

Answer 1

The Correct Option is C

Step 1: Chain rule.
$\dfrac{dV}{dx} = \dfrac{dV}{dr}\cdot\dfrac{dr}{dx} = 4\pi r^2 \cdot \dfrac{dr}{dx}$.

Step 2: Find the pieces.
$r = \dfrac{3x}{2} + \dfrac{9}{4}$, so $\dfrac{dr}{dx} = \dfrac{3}{2}$. Also $r^2 = \dfrac{9}{16}(2x + 3)^2$.

Step 3: Multiply.
\[ \frac{dV}{dx} = 4\pi\cdot\frac{9}{16}(2x+3)^2\cdot\frac{3}{2} = \frac{27\pi}{8}(2x+3)^2 \]

Final Answer:
Option (C). \[ \boxed{\frac{27\pi}{8}(2x+3)^2} \]