Question

An ice cream shop currently sells soft-serve ice cream in cones that have a radius of 3 inches and a height of 6 inches. The shop wants to offer a wider cone for their hard-serve ice cream that has the same volume as their existing cone but a wider radius of 4 inches. What will be the height, to the nearest inch, of the new cone?(1 point)
Responses

14 inches
14 inches

18 inches
18 inches

4 inches
4 inches

3 inches

Answer 1

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**