If a spherical balloon has a variable diameter `(3x+9//2)`., then the rate of change of its volume w. r. t x is

Options

A. `27 pi(2x+3)^(2)`
B. `(27pi)/(8) (2x+3)^(2)`
C. `(27pi)/(26)(2x+3)^(2)`
D. `pi (2x+3)^(3)`

Answer

To find the rate of change of the volume of a spherical balloon with respect to the variable \( x \), we will follow these steps:

### Step 1: Determine the Diameter and Radius
The diameter \( D \) of the spherical balloon is given as:
\[ D = 3x + \frac{9}{2} \]
The radius \( r \) is half of the diameter:
\[ r = \frac{D}{2} = \frac{3x + \frac{9}{2}}{2} = \frac{3x}{2} + \frac{9}{4} \]

### Step 2: Write the Volume Formula
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the expression for \( r \):
\[ V = \frac{4}{3} \pi \left(\frac{3x}{2} + \frac{9}{4}\right)^3 \]

### Step 3: Simplify the Volume Expression
Let’s simplify \( r \):
\[ r = \frac{3x + \frac{9}{2}}{2} = \frac{3x + 4.5}{2} \]
Now, we will calculate \( r^3 \):
\[ r^3 = \left(\frac{3x + 4.5}{2}\right)^3 = \frac{(3x + 4.5)^3}{8} \]
Thus, the volume becomes:
\[ V = \frac{4}{3} \pi \cdot \frac{(3x + 4.5)^3}{8} = \frac{\pi (3x + 4.5)^3}{6} \]

### Step 4: Differentiate the Volume with Respect to \( x \)
To find the rate of change of volume with respect to \( x \), we differentiate \( V \):
\[ \frac{dV}{dx} = \frac{\pi}{6} \cdot 3(3x + 4.5)^2 \cdot \frac{d}{dx}(3x + 4.5) \]
Since \( \frac{d}{dx}(3x + 4.5) = 3 \), we have:
\[ \frac{dV}{dx} = \frac{\pi}{6} \cdot 3(3x + 4.5)^2 \cdot 3 = \frac{9\pi}{6} (3x + 4.5)^2 = \frac{3\pi}{2} (3x + 4.5)^2 \]

### Final Result
Thus, the rate of change of the volume of the spherical balloon with respect to \( x \) is:
\[ \frac{dV}{dx} = \frac{3\pi}{2} (3x + 4.5)^2 \]