To find the height of the new cone that has the same volume as the existing cone but a wider radius, we first need to calculate the volume of the existing cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the radius of the cone,
- \( h \) is the height of the cone.
For the existing soft-serve ice cream cone:
- Radius (\( r \)) = 3 inches
- Height (\( h \)) = 6 inches
Calculating the volume of the existing cone:
\[ V = \frac{1}{3} \pi (3^2) (6) = \frac{1}{3} \pi (9) (6) = \frac{1}{3} \pi (54) = 18\pi \]
Now, we want to find the height of the new cone with a radius of 4 inches that has the same volume of \( 18\pi \).
Let \( h' \) be the height of the new cone. Using the volume formula:
\[ V = \frac{1}{3} \pi (r')^2 h' \]
Where \( r' = 4 \) inches, and we set this equal to the volume of the existing cone:
\[ 18\pi = \frac{1}{3} \pi (4^2) h' \]
Simplifying this equation:
\[ 18\pi = \frac{1}{3} \pi (16) h' \]
Cancel \( \pi \) from both sides:
\[ 18 = \frac{16}{3} h' \]
To find \( h' \), multiply both sides by \( \frac{3}{16} \):
\[ h' = 18 \cdot \frac{3}{16} = \frac{54}{16} = \frac{27}{8} \]
Now, convert \( \frac{27}{8} \) into a decimal:
\[ \frac{27}{8} = 3.375 \]
The nearest inch to \( 3.375 \) is \( 3 \) inches.
Therefore, the height of the new cone is 3 inches.