suppose a sphere cylinder and cone all share the same radius and the cylinder and cone also have 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?
answers: A: Sphere - cone = cylinder
B: sphere = cylinder + cone
C: 3 cones = 1 cylinder
Or D: 1 cylinder = 2 cones + 1 sphere.
The correct answer is:
D: 1 cylinder = 2 cones + 1 sphere.
To understand why this is the correct answer, we need to compare the volume formulas for each shape.
The volume of a sphere is given by
V_sphere = (4/3)πr^3,
The volume of a cylinder is given by
V_cylinder = πr^2h,
The volume of a cone is given by
V_cone = (1/3)πr^2h.
Given that the radius (r) is the same for all three shapes and the height (h) of the cylinder and cone is twice the radius, we can substitute these values into the volume formulas.
For the cylinder, with radius r and height 2r, we have
V_cylinder = πr^2(2r) = 2πr^3.
For the cone, with radius r and height 2r, we have
V_cone = (1/3)πr^2(2r) = (2/3)πr^3.
Adding the volume of the cone and the volume of the cylinder together, we get
(2/3)πr^3 + 2πr^3 = (8/3)πr^3.
Finally, if we compare this to the volume of the sphere, which is (4/3)πr^3, we find that
(4/3)πr^3 = (8/3)πr^3.
Therefore, it is true that 1 cylinder equals 2 cones plus 1 sphere.