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

3 inches
3 inches

14 inches
14 inches

4 inches
4 inches

18 inches
18 inches

Answer 1

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