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?(1 point)
A. sphere= cylinder+cone
B. sphere - cone=cylinder
C. 1 cylinder= 2 cones + 1 sphere
D. 3 cones= 1 cylinder
To solve this problem, we need to find the formulas for the volume of the sphere, cylinder, and cone and compare them.
The volume of a sphere with radius r is given by the formula:
V = (4/3)πr^3
The volume of a cylinder with radius r and height h is given by the formula:
V = πr^2h
The volume of a cone with radius r and height h is given by the formula:
V = (1/3)πr^2h
Since the radius is the same for all three shapes, we can rewrite the volume formulas in terms of the height h:
For the sphere: V_sphere = (4/3)πr^3 = (4/3)π(r^3/r^3)h^3 = (4/3)πh^3
For the cylinder: V_cylinder = πr^2h = π(r^2/r^2)h^3 = πh^3
For the cone: V_cone = (1/3)πr^2h = (1/3)π(r^2/r^2)h^3 = (1/3)πh^3
Comparing the formulas, we can see that the volume of the sphere is not equal to the sum of the volume of the cylinder and cone (A is false). The volume of the cylinder is not equal to the difference between the volume of the sphere and cone (B is false). The volume of the cylinder is not equal to the sum of 2 cones and 1 sphere (C is false). The volume of 3 cones is not equal to the volume of 1 cylinder (D is false).
Therefore, none of the given options are true about the relationship between the volume formulas for the sphere, cylinder, and cone.