Question
A sphere and a cylinder have the same radius and height. The volume of the cylinder is 50 feet cubed.
A cylinder with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
StartFraction 50 Over 3 EndFraction feet cubed
StartFraction 100 Over 3 EndFraction feet cubed
75 feet cubed
100 feet cubed
A cylinder with height h and radius r. A cylinder with height h and radius r.
What is the volume of the sphere?
StartFraction 50 Over 3 EndFraction feet cubed
StartFraction 100 Over 3 EndFraction feet cubed
75 feet cubed
100 feet cubed
Answers
GPT-4o mini
To find the volume of the sphere given that the cylinder has the same radius and height, we can use the formulas for the volume of the cylinder and the volume of the sphere.
**Volume of a cylinder (V_cylinder)**:
\[
V_{cylinder} = \pi r^2 h
\]
**Volume of a sphere (V_sphere)**:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
Given that the volume of the cylinder is 50 cubic feet, we can set up the equation:
\[
\pi r^2 h = 50
\]
Since the sphere and the cylinder have the same radius and height, we can express \(h\) in terms of \(r\). For the cylinder and sphere, the formulas can be related.
From the volume of the cylinder:
\[
h = \frac{50}{\pi r^2}
\]
Now, substitute \(h\) into the volume of the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
To express \(V_{sphere}\) directly in terms of the volume of the cylinder, we note that:
1. **From the volume of the cylinder**:
\( \pi r^2 h = 50 \).
So we can express \(r^2 h\) as:
\[
r^2 h = \frac{50}{\pi}
\]
2. Now, rewrite \(V_{sphere}\):
The radius \(r\) can be factored out from the relation we have, keeping in mind that from the cylinder volume, \(h\) cancels out.
Use the relationship \(h = \frac{50}{\pi r^2}\):
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
But to find \(V_{sphere}\) we also observe:
The volume relationship can also be understood by using the right proportions. Notice:
- \(V_{sphere} = \frac{4}{3} \cdot V_{cylinder} \cdot \frac{r}{h}\), where \(h=r\).
After simple calculations:
We can derive that the ratio \(V_{sphere}/V_{cylinder}\) implies approximate height equivalence, leading to:
\[
V_{sphere} = \frac{2}{3} \cdot 50 = \frac{100}{3}
\]
Thus, the volume of the sphere is:
\[
\frac{100}{3} \text{ cubic feet}
\]
The answer is:
\[
\boxed{\frac{100}{3} \text{ feet cubed}}
\]
**Volume of a cylinder (V_cylinder)**:
\[
V_{cylinder} = \pi r^2 h
\]
**Volume of a sphere (V_sphere)**:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
Given that the volume of the cylinder is 50 cubic feet, we can set up the equation:
\[
\pi r^2 h = 50
\]
Since the sphere and the cylinder have the same radius and height, we can express \(h\) in terms of \(r\). For the cylinder and sphere, the formulas can be related.
From the volume of the cylinder:
\[
h = \frac{50}{\pi r^2}
\]
Now, substitute \(h\) into the volume of the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
To express \(V_{sphere}\) directly in terms of the volume of the cylinder, we note that:
1. **From the volume of the cylinder**:
\( \pi r^2 h = 50 \).
So we can express \(r^2 h\) as:
\[
r^2 h = \frac{50}{\pi}
\]
2. Now, rewrite \(V_{sphere}\):
The radius \(r\) can be factored out from the relation we have, keeping in mind that from the cylinder volume, \(h\) cancels out.
Use the relationship \(h = \frac{50}{\pi r^2}\):
\[
V_{sphere} = \frac{4}{3} \pi r^3
\]
But to find \(V_{sphere}\) we also observe:
The volume relationship can also be understood by using the right proportions. Notice:
- \(V_{sphere} = \frac{4}{3} \cdot V_{cylinder} \cdot \frac{r}{h}\), where \(h=r\).
After simple calculations:
We can derive that the ratio \(V_{sphere}/V_{cylinder}\) implies approximate height equivalence, leading to:
\[
V_{sphere} = \frac{2}{3} \cdot 50 = \frac{100}{3}
\]
Thus, the volume of the sphere is:
\[
\frac{100}{3} \text{ cubic feet}
\]
The answer is:
\[
\boxed{\frac{100}{3} \text{ feet cubed}}
\]