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 inches, of the new cone
Answers
GPT-4o mini
To find the height of the new cone, we first need to calculate the volume of the existing soft-serve ice cream cone.
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cone.
For the existing cone, the radius \( r \) is 3 inches and the height \( h \) is 6 inches.
Substituting these values into the volume formula:
\[
V = \frac{1}{3} \pi (3^2)(6)
\]
\[
V = \frac{1}{3} \pi (9)(6)
\]
\[
V = \frac{1}{3} \pi (54)
\]
\[
V = 18\pi \text{ cubic inches}
\]
Now, we want to find the height of the new cone with a radius of 4 inches that has the same volume of \( 18\pi \) cubic inches.
Let the height of the new cone be \( h \). We will use the same volume formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting the radius of the new cone \( r = 4 \) inches and setting the volume equal to \( 18\pi \):
\[
18\pi = \frac{1}{3} \pi (4^2) h
\]
Now calculate \( (4^2) \):
\[
(4^2) = 16
\]
Substituting this back into the equation:
\[
18\pi = \frac{1}{3} \pi (16) h
\]
We can cancel \( \pi \) from both sides:
\[
18 = \frac{1}{3} (16) h
\]
To eliminate the fraction, multiply both sides by 3:
\[
54 = 16h
\]
Now, solve for \( h \):
\[
h = \frac{54}{16} = 3.375
\]
Rounding to the nearest inch gives:
\[
h \approx 3 \text{ inches}
\]
Thus, the height of the new cone is \( \boxed{3} \) inches.
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cone.
For the existing cone, the radius \( r \) is 3 inches and the height \( h \) is 6 inches.
Substituting these values into the volume formula:
\[
V = \frac{1}{3} \pi (3^2)(6)
\]
\[
V = \frac{1}{3} \pi (9)(6)
\]
\[
V = \frac{1}{3} \pi (54)
\]
\[
V = 18\pi \text{ cubic inches}
\]
Now, we want to find the height of the new cone with a radius of 4 inches that has the same volume of \( 18\pi \) cubic inches.
Let the height of the new cone be \( h \). We will use the same volume formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting the radius of the new cone \( r = 4 \) inches and setting the volume equal to \( 18\pi \):
\[
18\pi = \frac{1}{3} \pi (4^2) h
\]
Now calculate \( (4^2) \):
\[
(4^2) = 16
\]
Substituting this back into the equation:
\[
18\pi = \frac{1}{3} \pi (16) h
\]
We can cancel \( \pi \) from both sides:
\[
18 = \frac{1}{3} (16) h
\]
To eliminate the fraction, multiply both sides by 3:
\[
54 = 16h
\]
Now, solve for \( h \):
\[
h = \frac{54}{16} = 3.375
\]
Rounding to the nearest inch gives:
\[
h \approx 3 \text{ inches}
\]
Thus, the height of the new cone is \( \boxed{3} \) inches.