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)
• 54 cm
O 18 cm°
• 24 cm°
• 12 cm'
To solve this problem, we can use the relationship between the formulas for the volumes of a cylinder and a sphere.
The volume of a cylinder is given by the formula:
Vcylinder = π * r^2 * h,
where r is the radius and h is the height of the cylinder.
The volume of a sphere is given by the formula:
Vsphere = (4/3) * π * r^3,
where r is the radius of the sphere.
In this problem, we are told that the volume of the cylinder is 36 cm^3 and the height of the cylinder is equal to the sphere's diameter, which means h = 2r.
Substituting these values into the volume formula for the cylinder, we get:
36 = π * r^2 * 2r
Simplifying this equation, we get:
36 = 2π * r^3
Dividing both sides of the equation by 2π, we get:
18 = r^3
Taking the cube root of both sides of the equation, we get:
r = 3
Now that we have the radius of the sphere, we can substitute it into the volume formula for a sphere to find the volume:
Vsphere = (4/3) * π * (3)^3
Vsphere = (4/3) * π * 27
Vsphere = 36π
Therefore, the volume of the sphere with the same radius as the cylinder and a height equal to the sphere's diameter is 36π cm^3.
pick one of the options
The correct option is:
• 36π cm^3
no it is not
I apologize for the mistake. Let's solve the problem again.
We are given that the volume of the cylinder is 36 cm³. Using the formula for the volume of a cylinder:
Vcylinder = π * r^2 * h
And we know that the height of the cylinder is equal to the sphere's diameter, so h = 2r.
Substituting the values into the equation, we have:
36 = π * r^2 * 2r
Simplifying, we get:
36 = 2πr^3
Dividing both sides by 2π, we have:
18 = r^3
Taking the cube root of both sides, we find:
r = 3
Now that we have the radius of the cylinder, we can find the volume of the sphere using the formula for the volume of a sphere:
Vsphere = (4/3) * π * r^3
Substituting the value of r, we have:
Vsphere = (4/3) * π * 3^3
Vsphere = 36π
Therefore, the volume of the sphere is 36π cm³.