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

Answer 1

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 2

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 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!