A cylinder has a volume of `72pi` cubic inches and a height of 8 inches. If the height is increased by 4 inches, what will be the new volume of the cylinder in cubic inches?

Options

A. `76pi`
B. `108pi`
C. `328pi`
D. `576pi`

Answer

To find the new volume of the cylinder after increasing its height, we can follow these steps:

### Step 1: Understand the given information
We are given:
- Original volume of the cylinder, \( V = 72\pi \) cubic inches
- Original height of the cylinder, \( h = 8 \) inches

### Step 2: Calculate the radius of the original cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where \( r \) is the radius and \( h \) is the height. We can rearrange this formula to find the radius:
\[ r^2 = \frac{V}{\pi h} \]
Substituting the known values:
\[ r^2 = \frac{72\pi}{\pi \cdot 8} = \frac{72}{8} = 9 \]
Taking the square root:
\[ r = 3 \text{ inches} \]

### Step 3: Calculate the new height of the cylinder
The height is increased by 4 inches:
\[ \text{New height} = 8 + 4 = 12 \text{ inches} \]

### Step 4: Calculate the new volume of the cylinder
Now we can use the volume formula again with the new height:
\[ V_{\text{new}} = \pi r^2 h_{\text{new}} \]
Substituting the radius and the new height:
\[ V_{\text{new}} = \pi (3^2)(12) = \pi (9)(12) = 108\pi \text{ cubic inches} \]

### Final Answer
The new volume of the cylinder is \( 108\pi \) cubic inches.

---