Question
The volume of a sphere is 72 cm3 . What is the volume of a cone that shares the same radius of the sphere and its height is equal to the spheres diameter
Answers
GPT-4o mini
To solve this problem, we need to first determine the radius of the sphere from its volume and then use that radius to find the volume of the cone.
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Given that the volume of the sphere is 72 cm³, we can set up the equation:
\[
\frac{4}{3} \pi r^3 = 72
\]
To find the radius, we first isolate \( r^3 \):
\[
r^3 = \frac{72 \cdot 3}{4\pi}
\]
\[
r^3 = \frac{216}{4\pi}
\]
\[
r^3 = \frac{54}{\pi}
\]
\[
r = \left( \frac{54}{\pi} \right)^{\frac{1}{3}}
\]
Next, the diameter \( d \) of the sphere is:
\[
d = 2r
\]
The height \( h \) of the cone is equal to the diameter of the sphere:
\[
h = d = 2r
\]
Now we can find the volume \( V_{\text{cone}} \) of the cone, which is given by the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Substituting for \( h \):
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now we can substitute \( r^3 \) using the earlier calculation:
\[
V_{\text{cone}} = \frac{2}{3} \pi \left( \frac{54}{\pi} \right)
\]
Simplifying this gives:
\[
V_{\text{cone}} = \frac{2}{3} \cdot 54 = 36 \text{ cm}^3
\]
Thus, the volume of the cone is:
\[
\boxed{36 \text{ cm}^3}
\]
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Given that the volume of the sphere is 72 cm³, we can set up the equation:
\[
\frac{4}{3} \pi r^3 = 72
\]
To find the radius, we first isolate \( r^3 \):
\[
r^3 = \frac{72 \cdot 3}{4\pi}
\]
\[
r^3 = \frac{216}{4\pi}
\]
\[
r^3 = \frac{54}{\pi}
\]
\[
r = \left( \frac{54}{\pi} \right)^{\frac{1}{3}}
\]
Next, the diameter \( d \) of the sphere is:
\[
d = 2r
\]
The height \( h \) of the cone is equal to the diameter of the sphere:
\[
h = d = 2r
\]
Now we can find the volume \( V_{\text{cone}} \) of the cone, which is given by the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Substituting for \( h \):
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Now we can substitute \( r^3 \) using the earlier calculation:
\[
V_{\text{cone}} = \frac{2}{3} \pi \left( \frac{54}{\pi} \right)
\]
Simplifying this gives:
\[
V_{\text{cone}} = \frac{2}{3} \cdot 54 = 36 \text{ cm}^3
\]
Thus, the volume of the cone is:
\[
\boxed{36 \text{ cm}^3}
\]