Determine the volume of a cone with a circular base of area `9pi` and a height of 7x

Options

A. `3/7pix`
B. `21pix`
C. `9pixx7x`
D. `63pix`

Answer

To determine the volume of a cone with a circular base of area \(9\pi\) and a height of \(7x\), we can follow these steps:

### Step 1: Find the radius of the circular base
We know that the area \(A\) of a circle is given by the formula:
\[ A = \pi r^2 \]
Given that the area of the base is \(9\pi\), we can set up the equation:
\[ \pi r^2 = 9\pi \]
Now, we can divide both sides by \(\pi\):
\[ r^2 = 9 \]
Taking the square root of both sides gives:
\[ r = \sqrt{9} = 3 \]
Since the radius cannot be negative, we take \(r = 3\).

### Step 2: Use the volume formula for a cone
The formula for the volume \(V\) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
We have already found \(r = 3\) and the height \(h = 7x\). Plugging these values into the volume formula gives:
\[ V = \frac{1}{3} \pi (3^2) (7x) \]

### Step 3: Simplify the expression
Calculating \(3^2\):
\[ 3^2 = 9 \]
Now substituting this back into the volume formula:
\[ V = \frac{1}{3} \pi (9) (7x) \]
This simplifies to:
\[ V = \frac{1}{3} \cdot 9 \cdot 7\pi x \]

### Step 4: Calculate the final volume
Now, we can simplify further:
\[ V = \frac{63\pi x}{3} = 21\pi x \]

### Final Answer
Thus, the volume of the cone is:
\[ V = 21\pi x \text{ cubic units} \]