Question

suppose a sphere, cylinder, and a cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius? Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?
a 3 cones = 1 cylinder
b sphere - cone = cylinder
c 1 cylinder = 2 cones + 1 sphere
d sphere = cylinder + cone

Answer 1

The correct answer is d) sphere = cylinder + cone.

To understand why this is the case, let's look at the volume formulas for each shape.

The volume of a sphere is given by V = (4/3)πr^3, where r is the radius.

The volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height.

The volume of a cone is given by V = (1/3)πr^2h, where r is the radius and h is the height.

Given that the height of the cylinder and cone is twice the radius, h = 2r.

Substituting this value of h into the formulas, we get:

Volume of sphere = (4/3)πr^3
Volume of cylinder = πr^2(2r) = 2πr^3
Volume of cone = (1/3)πr^2(2r) = (2/3)πr^3

Now, if we add the volume of the cylinder and cone, we get:

Volume of cylinder + Volume of cone = 2πr^3 + (2/3)πr^3 = (8/3)πr^3

Comparing this with the volume of the sphere, we can see that:

(8/3)πr^3 = (4/3)πr^3

Thus, we can conclude that sphere = cylinder + cone, which matches option d).