Question

5. Volume of sphere is 60 cubic meters. Find volume of cylinder with same dimensions.

Answer 1

as soon as you determined that V = 2πr^3, you could just go on without having to find r.
(2πr^3) / (4/3 πr^3) = 2/(4/3) = 3/2
and 3/2 * 60 m^3 = 90 m^3

Answer 2

To find the volume of a cylinder that has the same dimensions as a sphere with a given volume, we need to first find the radius of the sphere and then use that radius to find the dimensions of the cylinder.

1. **Find the radius of the sphere:**
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
We know the volume of the sphere is 60 cubic meters:
\[
60 = \frac{4}{3} \pi r^3
\]
To solve for \( r^3 \), we can rearrange the equation:
\[
r^3 = \frac{60 \cdot 3}{4 \pi}
\]
\[
r^3 = \frac{180}{4 \pi} = \frac{45}{\pi}
\]
Now, taking the cube root to find \( r \):
\[
r = \sqrt[3]{\frac{45}{\pi}}
\]

2. **Find the volume of the cylinder:**
Assuming the cylinder has the same radius \( r \) and height equal to the diameter of the sphere (which is \( 2r \)), the volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
Here, the height \( h = 2r \), so:
\[
V = \pi r^2 (2r) = 2\pi r^3
\]
We already found \( r^3 = \frac{45}{\pi} \), so we can substitute it:
\[
V = 2\pi \left(\frac{45}{\pi}\right) = 2 \cdot 45 = 90 \text{ cubic meters}
\]

Thus, the volume of the cylinder with the same dimensions as the sphere is **90 cubic meters**.

Answer 3

You're absolutely correct! Given that the volumes relate directly through the proportionality constant, we can skip finding the radius altogether. Your method is much more efficient.

To summarize:

1. The volume of the sphere is given by
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
and we know that \( V_{\text{sphere}} = 60 \, \text{m}^3 \).

2. The volume of the cylinder is
\[
V_{\text{cylinder}} = 2 \pi r^3.
\]

3. To find the ratio of the volumes:
\[
\frac{V_{\text{cylinder}}}{V_{\text{sphere}}} = \frac{2\pi r^3}{\frac{4}{3} \pi r^3} = \frac{2}{\frac{4}{3}} = \frac{2 \cdot 3}{4} = \frac{3}{2}.
\]

4. Thus,
\[
V_{\text{cylinder}} = \frac{3}{2} \times V_{\text{sphere}} = \frac{3}{2} \times 60 \, \text{m}^3 = 90 \, \text{m}^3.
\]

So indeed, the volume of the cylinder is \( 90 \, \text{m}^3 \). Thank you for pointing out the more straightforward approach!