Question 1: Suppose a sphere, cylinder, and cone all share the same radius. 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?
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius.
The volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
In this case, the radius is the same for all three shapes, and the height of the cylinder and cone is twice the radius. Therefore, we can express the height of the cylinder and cone as h = 2r.
Plugging this value of h into the volume formulas for the cylinder and cone, we get:
V_cylinder = πr^2(2r) = 2πr^3
V_cone = (1/3)πr^2(2r) = (2/3)πr^3
Comparing the volume formulas, we can see that the volume of the cylinder is twice the volume of the sphere, while the volume of the cone is (2/3) times the volume of the sphere.
Therefore, the relationship between the volume formulas for the sphere, cylinder, and cone is:
V_cylinder = 2V_sphere
V_cone = (2/3)V_sphere