Question

the cylinder and the sphere below have the same radius and the same volume which is 6m

Answer 1

To find the radius of the cylinder and the sphere, we'll use the formulas for the volume of each shape.

1. **Volume of a cylinder**:
The volume \( V \) of a cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.

2. **Volume of a sphere**:
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]

Since both shapes have the same volume of 6 m³, we can set up the equations. Let's equate the volumes:

### For the Cylinder
Assuming the height of the cylinder is \( h \):
\[
\pi r^2 h = 6
\]

### For the Sphere
Using the volume formula for the sphere:
\[
\frac{4}{3} \pi r^3 = 6
\]

### Solving for Radius (r)

**From the Sphere's volume:**
\[
\frac{4}{3} \pi r^3 = 6
\]
Multiply both sides by \( \frac{3}{4\pi} \):
\[
r^3 = \frac{6 \times 3}{4\pi} = \frac{18}{4\pi} = \frac{9}{2\pi}
\]
Now take the cube root:
\[
r = \left(\frac{9}{2\pi}\right)^{\frac{1}{3}}
\]

### Finding the height of the Cylinder:

Substituting \( r \) back into the cylinder’s volume equation:
\[
\pi r^2 h = 6
\]
First, calculate \( r^2 \):
\[
r^2 = \left(\frac{9}{2\pi}\right)^{\frac{2}{3}} = \frac{9^{\frac{2}{3}}}{(2\pi)^{\frac{2}{3}}}
\]
Now substitute \( r^2 \) into the cylinder's volume equation:
\[
\pi \frac{9^{\frac{2}{3}}}{(2\pi)^{\frac{2}{3}}} h = 6
\]
This simplifies to:
\[
\frac{9^{\frac{2}{3}} h}{(2\pi)^{\frac{2}{3}}} = 6
\]
Now solve for \( h \):
\[
h = 6 \frac{(2\pi)^{\frac{2}{3}}}{9^{\frac{2}{3}}}
\]
You can compute \( r \) and \( h \) using a calculator for numerical approximations if needed.

This gives a general method for finding the radius of the shapes. If you need the specific numeric values or further assistance with calculations, let me know!