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
Responses
14 inches
14 inches
18 inches
18 inches
4 inches
4 inches
3 inches
Answers
GPT-4o mini
To find the height of the new cone with a radius of 4 inches that has the same volume as the existing cone, we start by calculating the volume of the original cone.
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
For the original cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches
Substituting these values into the volume formula:
\[
V = \frac{1}{3} \pi (3)^2 (6)
\]
Calculating:
\[
V = \frac{1}{3} \pi (9)(6) = \frac{1}{3} \pi (54) = 18\pi \text{ cubic inches}
\]
Now, we want the volume of the new cone with a radius of 4 inches. Let's denote the height of the new cone as \( h \).
Using the same volume formula:
\[
V = \frac{1}{3} \pi (4)^2 h
\]
Substituting the values:
\[
V = \frac{1}{3} \pi (16) h = \frac{16}{3} \pi h
\]
We want this volume to equal the original volume \( 18\pi \):
\[
\frac{16}{3} \pi h = 18\pi
\]
Dividing both sides by \( \pi \):
\[
\frac{16}{3} h = 18
\]
Now, multiply both sides by \( \frac{3}{16} \) to 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 \( 3.375 \) to the nearest inch gives us \( 3 \) inches.
Thus, the height of the new cone is **3 inches**.
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
For the original cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches
Substituting these values into the volume formula:
\[
V = \frac{1}{3} \pi (3)^2 (6)
\]
Calculating:
\[
V = \frac{1}{3} \pi (9)(6) = \frac{1}{3} \pi (54) = 18\pi \text{ cubic inches}
\]
Now, we want the volume of the new cone with a radius of 4 inches. Let's denote the height of the new cone as \( h \).
Using the same volume formula:
\[
V = \frac{1}{3} \pi (4)^2 h
\]
Substituting the values:
\[
V = \frac{1}{3} \pi (16) h = \frac{16}{3} \pi h
\]
We want this volume to equal the original volume \( 18\pi \):
\[
\frac{16}{3} \pi h = 18\pi
\]
Dividing both sides by \( \pi \):
\[
\frac{16}{3} h = 18
\]
Now, multiply both sides by \( \frac{3}{16} \) to 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 \( 3.375 \) to the nearest inch gives us \( 3 \) inches.
Thus, the height of the new cone is **3 inches**.