Question

page 16 of 16 Volume of a Cone Quick Check 3 of 53 of 5 Items 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 3 inches 3 inches 4 inches 4 inches 18 inches 18 inches Skip to navigation

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 height of 6 inches, we start by calculating the volume of the original cone.

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.

### Step 1: Calculate the volume of the existing cone

For the existing cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches

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

### Step 2: Set the volume of the new cone equal to the existing cone's volume

For the new cone:
- Radius \( r = 4 \) inches
- Height \( h \) (unknown)

Using the volume formula:

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

### Step 3: Set the volumes equal to each other

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

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

Divide both sides by \( \pi \):

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

Multiply both sides by \( \frac{3}{16} \):

\[
h = 18 \cdot \frac{3}{16}
\]
\[
= \frac{54}{16}
\]
\[
= 3.375
\]

### Step 5: Round to the nearest inch

The height \( h \) of the new cone, rounding to the nearest inch, is:

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

Therefore, the correct answer is **3 inches**.