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

4 inches
4 inches

18 inches
18 inches

14 inches
14 inches

3 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 determine the volume of the existing cone.

The volume \( V \) of a cone is calculated using the formula:

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

where \( r \) is the radius and \( h \) is the height.

### Step 1: Calculate the volume of the existing cone
For the existing cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches

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

### Step 2: Set the volume of the new cone equal to the existing cone's volume
The new cone has a radius of 4 inches, and we need to find its height \( h' \).

\[
V' = \frac{1}{3} \pi (r')^2 h'
\]
Where \( r' = 4 \) inches.

Setting the volumes equal gives:

\[
18 \pi = \frac{1}{3} \pi (4^2) h'
\]

### Step 3: Simplify the equation
Cancel \( \pi \) from both sides:

\[
18 = \frac{1}{3} (16) h'
\]

### Step 4: Solve for \( h' \)

Multiply both sides by 3:

\[
54 = 16 h'
\]

Now, divide by 16:

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

### Conclusion: Round to the nearest inch

The height of the new cone is approximately \( 3.375 \) inches, which rounds to **3 inches**.

Thus, the answer is:

**3 inches**.