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)
sphere = cylinder + cone
• cone = sphere - cylinder
O cylinder = sphere - cone
• cone = cylinder - sphere
The correct answer is cone = sphere - cylinder.
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*
. 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? (1 point)
• 12 cm
• 54 cm°
• 24 cm3
0 18 cm
The volume of a cylinder is given by the formula V = πr^2h, and we are told that the volume of the cylinder is 36 cm^3. Let's assume the radius of the cylinder is r.
Given that the height of the cylinder is equal to the diameter of the sphere, which means the height of the cylinder is twice the radius. So, h = 2r.
Substituting this into the volume formula for the cylinder, we get:
36 = πr^2 * 2r
Simplifying this equation, we get:
36 = 2πr^3
Dividing both sides by 2π, we get:
r^3 = 18/π
Taking the cubic root of both sides, we get:
r = (18/π)^(1/3)
Now, we can calculate the volume of the sphere using the formula V = (4/3)πr^3:
V = (4/3)π * (18/π)^(1/3)^3
V = (4/3)π * 18/π
V = (4/3) * 18
V = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.
The volume of a cone is
cm'. What is the volume of a sphere if its radius is the same as the cone's and the
height of the cone is equal to the sphere's diameter? (1 point)
25
2
T cm
25
T cm
O 25 т ст°
50
3
The volume of a cone is given by the formula V = (1/3)πr^2h, and we are told that the volume of the cone is 25 cm^3. Let's assume the radius of the cone is r.
Given that the height of the cone is equal to the diameter of the sphere, which means the height of the cone is twice the radius. So, h = 2r.
Substituting this into the volume formula for the cone, we get:
25 = (1/3)πr^2 * 2r
Simplifying this equation, we get:
25 = (2/3)πr^3
Dividing both sides by (2/3)π, we get:
r^3 = 25 / ((2/3)π)
Now, we can calculate the volume of the sphere using the formula V = (4/3)πr^3:
V = (4/3)π * ((25 / ((2/3)π))^(1/3))^3
V = (4/3)π * (25 / ((2/3)π))
V = (4/3) * (25 / (2/3))
V = (4/3) * (25 * (3/2))
V = (4/3) * (75/2)
V = (4/3) * 37.5
V = 50 cm^3
Therefore, the volume of the sphere is 50 cm^3.