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
Responses
4 inches
4 inches
18 inches
18 inches
14 inches
14 inches
3 inches
Answers
GPT-4o mini
To find the height of the new cone, we first need to calculate the volume of the original soft-serve cone and then set it equal to the volume of the new cone with a wider radius.
The volume \( V \) of a cone can be 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 original cone.**
The original cone has a radius \( r = 3 \) inches and 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 volume of the old cone.**
The new cone has a radius \( r = 4 \) inches, and we want to find its height \( h \).
\[
V = \frac{1}{3} \pi (4)^2 h
\]
Setting the volumes equal to each other:
\[
\frac{1}{3} \pi (4)^2 h = 18\pi
\]
**Step 3: Simplify and solve for \( h \).**
\[
\frac{1}{3} \pi (16) h = 18\pi
\]
Divide both sides by \( \pi \):
\[
\frac{1}{3} (16) h = 18
\]
Multiply both sides by 3:
\[
16h = 54
\]
Now divide by 16:
\[
h = \frac{54}{16} = 3.375
\]
Since we need to provide the height in whole inches, we round \( 3.375 \) to the nearest inch, which gives us \( 3 \) inches.
**Final Answer: 3 inches.**
The volume \( V \) of a cone can be 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 original cone.**
The original cone has a radius \( r = 3 \) inches and 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 volume of the old cone.**
The new cone has a radius \( r = 4 \) inches, and we want to find its height \( h \).
\[
V = \frac{1}{3} \pi (4)^2 h
\]
Setting the volumes equal to each other:
\[
\frac{1}{3} \pi (4)^2 h = 18\pi
\]
**Step 3: Simplify and solve for \( h \).**
\[
\frac{1}{3} \pi (16) h = 18\pi
\]
Divide both sides by \( \pi \):
\[
\frac{1}{3} (16) h = 18
\]
Multiply both sides by 3:
\[
16h = 54
\]
Now divide by 16:
\[
h = \frac{54}{16} = 3.375
\]
Since we need to provide the height in whole inches, we round \( 3.375 \) to the nearest inch, which gives us \( 3 \) inches.
**Final Answer: 3 inches.**