Question

How is the formula for the volume of a sphere derived?

The sphere and the cylinder with two cones removed have equal volumes. The volume of the cylinder with radius (r) and height (2r) is (2pi r³). The volume of each cone with radius (r) and height (r) is (1/3pi r³). The volume of the cylinder with two cones removed is ______.

a) (3pi r³)

b) (4pi r³)

c) (2pi r³)

d) (5pi r³)

Answer 1

To find the volume of the cylinder with two cones removed, subtract the volume of the two cones from the cylinder's volume. The resulting volume is (4/3)πr³, which is also the volume of a sphere.

Step-by-step explanation:

The question is asking to calculate the volume of a cylinder with two cones removed. This can be solved using the given volume formulas for a cylinder and a cone. The volume of a cylinder is given by V = πr²h, and for a cone, it is V = (1/3)πr³. Given that the cylinder has a radius r and a height of 2r, its volume is 2πr³. Each cone has a radius r and a height r; thus, the volume of each cone is (1/3)πr³. To find the volume of the cylinder with two cones removed, we subtract the combined volume of the two cones from the volume of the cylinder:

Cylinder volume: 2πr³
Combined cones' volume: 2 × (1/3)πr³ = (2/3)πr³
Cylinder with two cones removed: 2πr³ - (2/3)πr³ = (4/3)πr³

So, the correct answer is b) (4πr³), which is the volume of the sphere.