Suppose a sphere, cylinder, and 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:1 cylinder = 2 cones + 1 sphere
C:sphere - cone = cylinder
D:sphere = cylinder + cone
To find the relationship between the volume formulas, we need to compare the volumes of the sphere, cylinder, and cone.
The volume of a sphere with radius r is given by V_sphere = 4/3 * π * r^3.
The volume of a cylinder with radius r and height h is given by V_cylinder = π * r^2 * h.
The volume of a cone with radius r and height h is given by V_cone = 1/3 * π * r^2 * h.
In this case, the radius and height of the cylinder and cone are both equal to r, and the height of the cylinder is twice the radius, so h = 2r.
Now let's substitute the value of h into the volume formulas for the cylinder and cone:
V_cylinder = π * r^2 * (2r) = 2π * r^3
V_cone = 1/3 * π * r^2 * (2r) = 2/3 * π * r^3
Comparing the volumes, we have:
V_sphere : V_cylinder : V_cone = 4/3 * π * r^3 : 2π * r^3 : 2/3 * π * r^3
To simplify the comparison, we can cancel out the common factor of r^3:
V_sphere : V_cylinder : V_cone = 4/3 : 2 : 2/3
Now let's find a relationship using the given options:
A: 3 cones = 1 cylinder
Comparing the ratios, 2/(2/3) ≠ 3, so this option is not true.
B: 1 cylinder = 2 cones + 1 sphere
Comparing the ratios, 2/(2/3) + 1 ≠ 4/3, so this option is not true.
C: sphere - cone = cylinder
Comparing the ratios, 4/3 - 2/3 ≠ 2, so this option is not true.
D: sphere = cylinder + cone
Comparing the ratios, 4/3 = 2 + 2/3, so this option is true.
Therefore, the correct answer is:
D: sphere = cylinder + cone.