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

18 inches
18 inches

3 inches
3 inches

4 inches
4 inches

14 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 with a radius of 3 inches and a height of 6 inches, we can use the formula for the volume of a cone:

\[
V = \frac{1}{3} \pi r^2 h
\]

**Step 1: Calculate the volume of the existing cone.**

For the original cone:

- Radius \( r_1 = 3 \) inches
- Height \( h_1 = 6 \) inches

Calculating the volume \( V_1 \):

\[
V_1 = \frac{1}{3} \pi (3)^2 (6)
\]
\[
V_1 = \frac{1}{3} \pi (9) (6)
\]
\[
V_1 = \frac{1}{3} \pi (54)
\]
\[
V_1 = 18\pi \text{ cubic inches}
\]

**Step 2: Set up the equation for the new cone.**

For the new cone:

- Radius \( r_2 = 4 \) inches
- Height \( h_2 \) (unknown)

The volume \( V_2 \) is given as:

\[
V_2 = \frac{1}{3} \pi (4)^2 h_2
\]
\[
V_2 = \frac{1}{3} \pi (16) h_2
\]
\[
V_2 = \frac{16}{3} \pi h_2
\]

**Step 3: Set the volumes equal to each other.**

Since both cones have the same volume:

\[
18\pi = \frac{16}{3} \pi h_2
\]

**Step 4: Solve for \( h_2 \).**

Divide both sides by \( \pi \):

\[
18 = \frac{16}{3} h_2
\]

Multiply both sides by 3 to eliminate the fraction:

\[
54 = 16 h_2
\]

Now, divide both sides by 16:

\[
h_2 = \frac{54}{16}
\]
\[
h_2 = 3.375 \text{ inches}
\]

Rounding to the nearest inch, we get:

\[
h_2 \approx 3 \text{ inches}
\]

**Conclusion:**
The height of the new cone will be **3 inches**.