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 calculate the volume of the original 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.
Volume of the Original Cone
For the original cone:
- Radius (\( r \)) = 3 inches
- Height (\( h \)) = 6 inches
Calculating the volume:
\[ V = \frac{1}{3} \pi (3^2)(6) = \frac{1}{3} \pi (9)(6) = \frac{1}{3} \pi (54) = 18\pi \text{ cubic inches} \]
Volume of the New Cone
For the new cone, we want it to have the same volume of \( 18\pi \) cubic inches and a radius of 4 inches.
Let the height of the new cone be \( H \). The volume of the new cone can be expressed as:
\[ V = \frac{1}{3} \pi (4^2) H = \frac{1}{3} \pi (16) H = \frac{16}{3} \pi H \]
Setting the Volumes Equal
We set the volume of the new cone equal to the volume of the original cone:
\[ \frac{16}{3} \pi H = 18\pi \]
To eliminate \( \pi \) from both sides, we divide each side by \( \pi \):
\[ \frac{16}{3} H = 18 \]
Solving for \( H \)
Now, we solve for \( H \):
\[ H = 18 \cdot \frac{3}{16} = \frac{54}{16} = \frac{27}{8} \]
Calculating \( \frac{27}{8} \):
\[ \frac{27}{8} = 3.375 \]
Rounding to the Nearest Inch
Since the question asks for the height to the nearest inch, we round \( 3.375 \) to \( 3 \).
Thus, the height of the new cone is:
3 inches