Give the dimensions of a cylinder and a sphere that have the same volume.
4/3 pi r^3 = pi r^2 h
h = 4/3 r
To find the dimensions of a cylinder and a sphere with the same volume, we can equate the volume formulas for both shapes and solve for the respective dimensions.
Let's assume:
- The height of the cylinder is represented by 'h'.
- The radius of the cylinder is represented by 'r'.
- The radius of the sphere is represented by 'R'.
The formulas for the volume of a cylinder and a sphere are as follows:
Volume of cylinder = π * r^2 * h
Volume of sphere = (4/3) * π * R^3
Since we want the volume to be equal, we can set up the equation:
π * r^2 * h = (4/3) * π * R^3
To find the dimensions, we need to eliminate one variable. We can choose to solve for 'h' in terms of 'R' or 'R' in terms of 'h'. Solving for 'h' in terms of 'R' is easier, so let's do that.
First, cancel out π on both sides of the equation:
r^2 * h = (4/3) * R^3
Next, divide both sides by r^2:
h = (4/3) * R^3 / r^2
Therefore, in order to have a cylinder and a sphere with the same volume, the height 'h' of the cylinder should be equal to (4/3) multiplied by the cube of the radius 'R' of the sphere, divided by the square of the radius 'r' of the cylinder. The radius 'R' of the sphere can be any value, while the radius 'r' of the cylinder can be any reasonable value as long as it is not zero.