Use the relationship among the formulas for the volumes of cones cylinders and Spears to solve the following problem the volume of a cylinder is 36 CM 3 what is the volume of a sphere if it's radius is the same as the cylinders and the height of the cylinder is equal to the Spheres them diameter
To find the volume of a sphere with the same radius as the given cylinder and a diameter equal to the height of the cylinder, we can use the formulas for the volumes of a cylinder and a sphere. Let's denote the radius as r and the height of the cylinder as h.
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
For a sphere, the formula for the volume is:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
You stated that the volume of the cylinder is 36 cm³. To find the volume of the corresponding sphere, we need to find the radius of the cylinder first. But we also have the additional information that the cylinder's height is equal to the sphere's diameter (2r), which gives us h = 2r. Let's substitute h = 2r into the cylinder's volume formula and solve for r:
\[ 36 = \pi r^2 (2r) \]
\[ 36 = 2\pi r^3 \]
\[ r^3 = \frac{36}{2\pi} \]
\[ r^3 = \frac{18}{\pi} \]
Now, let's use this value of r to find the volume of the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
\[ V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{18}{\pi}\right) \]
\[ V_{\text{sphere}} = \frac{4}{3} \cdot 18 \]
\[ V_{\text{sphere}} = 4 \cdot 6 \]
\[ V_{\text{sphere}} = 24 \text{ cm}^3 \]
Therefore, the volume of the sphere with the same radius as the cylinder and a diameter equal to the height of the cylinder is 24 cm³.