To find the height of the new cone with a radius of 4 inches that has the same volume as the existing cone, we first need to determine the volume of the existing cone.
The volume \( V \) of a cone is calculated using the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
Step 1: Calculate the volume of the existing cone
For the existing cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches
\[ V = \frac{1}{3} \pi (3^2)(6) \] \[ V = \frac{1}{3} \pi (9)(6) \] \[ V = \frac{1}{3} \pi (54) \] \[ V = 18 \pi \ \text{cubic inches} \]
Step 2: Set the volume of the new cone equal to the existing cone's volume
The new cone has a radius of 4 inches, and we need to find its height \( h' \).
\[ V' = \frac{1}{3} \pi (r')^2 h' \] Where \( r' = 4 \) inches.
Setting the volumes equal gives:
\[ 18 \pi = \frac{1}{3} \pi (4^2) h' \]
Step 3: Simplify the equation
Cancel \( \pi \) from both sides:
\[ 18 = \frac{1}{3} (16) h' \]
Step 4: Solve for \( h' \)
Multiply both sides by 3:
\[ 54 = 16 h' \]
Now, divide by 16:
\[ h' = \frac{54}{16} = 3.375 \ \text{inches} \]
Conclusion: Round to the nearest inch
The height of the new cone is approximately \( 3.375 \) inches, which rounds to 3 inches.
Thus, the answer is:
3 inches.