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
Responses
3 inches
3 inches
14 inches
14 inches
4 inches
4 inches
18 inches
Answers
GPT-4o mini
To find the height of the new cone that has a radius of 4 inches and the same volume as the original cone, we need to first calculate the volume of the original cone.
The formula for the volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
For the original cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches
Now, we can calculate the volume of the original cone:
\[
V = \frac{1}{3} \pi (3)^2 (6) = \frac{1}{3} \pi (9)(6) = \frac{54}{3} \pi = 18\pi \text{ cubic inches}
\]
Next, we want the volume of the new cone with a radius of 4 inches to be equal to this volume. Let \( h \) be the height of the new cone. Using the volume formula again for the new cone:
\[
V = \frac{1}{3} \pi (4)^2 h
\]
Setting this equal to the volume of the original cone:
\[
\frac{1}{3} \pi (4)^2 h = 18 \pi
\]
We can cancel \( \pi \) from both sides:
\[
\frac{1}{3} (4)^2 h = 18
\]
Calculating \( (4)^2 \):
\[
\frac{1}{3} (16) h = 18
\]
Multiplying both sides by 3:
\[
16h = 54
\]
Now, solving for \( h \):
\[
h = \frac{54}{16} = 3.375
\]
Rounding to the nearest inch, the height of the new cone is approximately 3 inches.
So the answer is:
**3 inches**
The formula for the volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
For the original cone:
- Radius \( r = 3 \) inches
- Height \( h = 6 \) inches
Now, we can calculate the volume of the original cone:
\[
V = \frac{1}{3} \pi (3)^2 (6) = \frac{1}{3} \pi (9)(6) = \frac{54}{3} \pi = 18\pi \text{ cubic inches}
\]
Next, we want the volume of the new cone with a radius of 4 inches to be equal to this volume. Let \( h \) be the height of the new cone. Using the volume formula again for the new cone:
\[
V = \frac{1}{3} \pi (4)^2 h
\]
Setting this equal to the volume of the original cone:
\[
\frac{1}{3} \pi (4)^2 h = 18 \pi
\]
We can cancel \( \pi \) from both sides:
\[
\frac{1}{3} (4)^2 h = 18
\]
Calculating \( (4)^2 \):
\[
\frac{1}{3} (16) h = 18
\]
Multiplying both sides by 3:
\[
16h = 54
\]
Now, solving for \( h \):
\[
h = \frac{54}{16} = 3.375
\]
Rounding to the nearest inch, the height of the new cone is approximately 3 inches.
So the answer is:
**3 inches**