To solve the problem, we first need to calculate the volume of the current soft-serve ice cream cone and then use that volume to determine the height of the new cone with a radius of 4 inches.
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 and \( h \) is the height of the cone.
Step 1: Calculate the volume of the current cone.
For the current soft-serve cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches
Now, substituting these values into the volume formula:
\[ V = \frac{1}{3} \pi (3)^2 (6) \]
Calculating:
\[ V = \frac{1}{3} \pi (9)(6) = \frac{1}{3} \pi (54) = 18\pi \]
The volume of the existing cone is \( 18\pi \) cubic inches.
Step 2: Determine the height of the new cone with a radius of 4 inches.
For the new cone:
- Radius \( r = 4 \) inches
- We will denote the height as \( h \).
Using the volume formula again:
\[ 18\pi = \frac{1}{3} \pi (4)^2 h \]
Calculating \( (4)^2 = 16 \):
\[ 18\pi = \frac{1}{3} \pi (16) h \]
Now simplify the right side:
\[ 18\pi = \frac{16}{3} \pi h \]
To eliminate \( \pi \), we can divide both sides by \( \pi \):
\[ 18 = \frac{16}{3} h \]
Next, we multiply both sides by 3 to isolate \( h \):
\[ 54 = 16h \]
Now, divide both sides by 16:
\[ h = \frac{54}{16} = 3.375 \]
Rounding this to the nearest inch gives us:
\[ h \approx 3 \text{ inches} \]
Thus, the height of the new cone, to the nearest inch, is 3 inches.
The answer is: 3 inches