If the height and the radius of a cone are doubled, the volume of the cone becomes

Options

A. 3 times
B. 4 times
C. 6 times
D. 8 times

Explanation

To solve the problem of how the volume of a cone changes when both its height and radius are doubled, we can follow these steps:

### Step 1: Understand the formula for the volume of a cone.
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

**Hint:** Remember that the volume of a cone depends on both the radius and the height.

### Step 2: Write down the original volume.
Let’s denote the original radius as \( r \) and the original height as \( h \). The original volume \( V \) can be expressed as:
\[ V = \frac{1}{3} \pi r^2 h \]

**Hint:** This is the starting point for our calculations.

### Step 3: Determine the new dimensions.
If the height and the radius of the cone are both doubled, then the new radius \( r' \) and the new height \( h' \) can be expressed as:
\[ r' = 2r \quad \text{and} \quad h' = 2h \]

**Hint:** Keep track of how the dimensions change when they are doubled.

### Step 4: Calculate the new volume.
Now, we can calculate the new volume \( V' \) using the new dimensions:
\[ V' = \frac{1}{3} \pi (r')^2 (h') \]
Substituting the new dimensions:
\[ V' = \frac{1}{3} \pi (2r)^2 (2h) \]
Calculating \( (2r)^2 \):
\[ (2r)^2 = 4r^2 \]
So, substituting this back into the volume formula gives:
\[ V' = \frac{1}{3} \pi (4r^2)(2h) \]
This simplifies to:
\[ V' = \frac{1}{3} \pi (8r^2 h) \]

**Hint:** Notice how the volume expression is changing as we substitute the new dimensions.

### Step 5: Relate the new volume to the original volume.
We can express the new volume in terms of the original volume \( V \):
\[ V' = 8 \left( \frac{1}{3} \pi r^2 h \right) = 8V \]

**Hint:** This shows that the new volume is a multiple of the original volume.

### Conclusion:
Thus, when both the height and the radius of the cone are doubled, the volume of the cone becomes 8 times the original volume.

**Final Answer:** The volume of the cone becomes \( 8V \), where \( V \) is the original volume of the cone.

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