Question
Volume of Cones, Cylinders, and Spheres Practice
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Question
The volume of a sphere is 72 m3
. What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
Answers
GPT-4o mini
To find the volume of the cone, we first need to determine the radius and diameter of the sphere from its volume.
The formula for the volume of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
where \( V \) is the volume and \( r \) is the radius.
Given that the volume of the sphere is 72 m³, we can set up the equation:
\[
\frac{4}{3} \pi r^3 = 72
\]
To solve for \( r \):
1. Multiply both sides by \( \frac{3}{4} \):
\[
\pi r^3 = 72 \cdot \frac{3}{4} = 54
\]
2. Divide by \( \pi \):
\[
r^3 = \frac{54}{\pi}
\]
3. Take the cube root of both sides to find \( r \):
\[
r = \sqrt[3]{\frac{54}{\pi}}
\]
Next, we need to calculate the diameter of the sphere, which is:
\[
d = 2r
\]
The formula for the volume of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( h \) is the height of the cone, and since the height is equal to the diameter of the sphere, we can substitute \( h \) with \( 2r \):
Thus, the volume of the cone becomes:
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now substituting \( r^3 \) from the earlier calculation:
\[
V = \frac{2}{3} \pi \left(\frac{54}{\pi}\right)
\]
Simplifying further:
\[
V = \frac{2 \times 54}{3} = \frac{108}{3} = 36
\]
Therefore, the volume of the cone is \( \boxed{36 \, m^3} \).
The formula for the volume of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
where \( V \) is the volume and \( r \) is the radius.
Given that the volume of the sphere is 72 m³, we can set up the equation:
\[
\frac{4}{3} \pi r^3 = 72
\]
To solve for \( r \):
1. Multiply both sides by \( \frac{3}{4} \):
\[
\pi r^3 = 72 \cdot \frac{3}{4} = 54
\]
2. Divide by \( \pi \):
\[
r^3 = \frac{54}{\pi}
\]
3. Take the cube root of both sides to find \( r \):
\[
r = \sqrt[3]{\frac{54}{\pi}}
\]
Next, we need to calculate the diameter of the sphere, which is:
\[
d = 2r
\]
The formula for the volume of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( h \) is the height of the cone, and since the height is equal to the diameter of the sphere, we can substitute \( h \) with \( 2r \):
Thus, the volume of the cone becomes:
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now substituting \( r^3 \) from the earlier calculation:
\[
V = \frac{2}{3} \pi \left(\frac{54}{\pi}\right)
\]
Simplifying further:
\[
V = \frac{2 \times 54}{3} = \frac{108}{3} = 36
\]
Therefore, the volume of the cone is \( \boxed{36 \, m^3} \).