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?
O sphere = cylinder + cone
O cone = cylinder – sphere
O cylinder = sphere – cone
O cone = sphere – cylinder
2. Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm^3
. What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?
O 24 cm^3
O 18 cm^3
O 12 cm^3
O 54 cm^3
1. O sphere = cylinder + cone
2. The volume of a cylinder is given by the formula V = πr^2h. If the volume of the cylinder is 36 cm^3 and the height of the cylinder is equal to the diameter of the sphere, we can set up the equation:
36 = πr^2(2r)
Simplifying the equation:
36 = 2πr^3
Dividing both sides by 2π:
18 = r^3
Taking the cube root of both sides to solve for r:
r = 3
Now we can use the volume formula for a sphere, V = (4/3)πr^3, to find the volume of the sphere:
V = (4/3)(π(3)^3)
V = (4/3)(π(27))
V = (4/3)(27π)
V = 36π
The volume of the sphere is 36π cm^3, which is equivalent to approximately 113.097 cm^3.
So the answer is none of the options provided.