To find the height of the new cone that has a wider radius but the same volume as the existing cone, we first need to calculate the volume of the original soft-serve ice cream 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 of the cone
- \( h \) is the height of the cone
For the original cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches
Calculating the volume of the original cone:
\[ V = \frac{1}{3} \pi (3)^2 (6) \] \[ V = \frac{1}{3} \pi (9)(6) \] \[ V = \frac{1}{3} \pi (54) \] \[ V = 18 \pi \text{ cubic inches} \]
Now we need to find the height of the new cone with a radius of 4 inches, which must have the same volume of \( 18 \pi \) cubic inches.
Let \( h' \) be the height of the new cone. We can set up the equation:
\[ V = \frac{1}{3} \pi (4)^2 h' \]
Setting this equal to the volume of the original cone:
\[ \frac{1}{3} \pi (4)^2 h' = 18 \pi \]
We can simplify this equation:
\[ \frac{1}{3} \pi (16) h' = 18 \pi \] \[ \frac{16}{3} h' = 18 \]
To solve for \( h' \), multiply both sides by \( \frac{3}{16} \):
\[ h' = 18 \times \frac{3}{16} \] \[ h' = \frac{54}{16} \] \[ h' = 3.375 \]
Rounding \( 3.375 \) to the nearest whole number gives us \( 3 \) inches. Thus, the height of the new cone is:
3 inches.