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 14 inches 14 inches 3 inches 3 inches 18 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 original cone, we first need to calculate 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 with a radius \( r = 3 \) inches and height \( h = 6 \) inches, the volume can be calculated as follows:
\[
V = \frac{1}{3} \pi (3)^2 (6)
\]
\[
V = \frac{1}{3} \pi (9)(6)
\]
\[
V = \frac{1}{3} \pi (54)
\]
\[
V = 18\pi \text{ cubic inches}
\]
Now, we know the volume of the new cone (with a radius of \( r = 4 \) inches) must also equal \( 18\pi \) cubic inches. We will use the volume formula again, this time solving for the height \( h \) of the new cone.
Let \( h' \) be the height of the new cone:
\[
V = \frac{1}{3} \pi (4)^2 h'
\]
\[
18\pi = \frac{1}{3} \pi (16) h'
\]
Now, we can cancel \( \pi \) from both sides:
\[
18 = \frac{1}{3} (16) h'
\]
To solve for \( h' \), first multiply both sides by 3:
\[
54 = 16 h'
\]
Now, divide both sides by 16:
\[
h' = \frac{54}{16} = 3.375 \text{ inches}
\]
Rounding to the nearest inch, the height \( h' \) is approximately \( 3 \) 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 with a radius \( r = 3 \) inches and height \( h = 6 \) inches, the volume can be calculated as follows:
\[
V = \frac{1}{3} \pi (3)^2 (6)
\]
\[
V = \frac{1}{3} \pi (9)(6)
\]
\[
V = \frac{1}{3} \pi (54)
\]
\[
V = 18\pi \text{ cubic inches}
\]
Now, we know the volume of the new cone (with a radius of \( r = 4 \) inches) must also equal \( 18\pi \) cubic inches. We will use the volume formula again, this time solving for the height \( h \) of the new cone.
Let \( h' \) be the height of the new cone:
\[
V = \frac{1}{3} \pi (4)^2 h'
\]
\[
18\pi = \frac{1}{3} \pi (16) h'
\]
Now, we can cancel \( \pi \) from both sides:
\[
18 = \frac{1}{3} (16) h'
\]
To solve for \( h' \), first multiply both sides by 3:
\[
54 = 16 h'
\]
Now, divide both sides by 16:
\[
h' = \frac{54}{16} = 3.375 \text{ inches}
\]
Rounding to the nearest inch, the height \( h' \) is approximately \( 3 \) inches.
Thus, the height of the new cone is:
**3 inches**.