The volume of a cone is 2/53 π cm^3 . 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?
Let's start by finding the volume of the cone.
The volume of a cone is given by the formula:
Vcone = (1/3)πr^2h
where r is the radius of the base and h is the height.
We are given that the volume of the cone is (2/53)π cm^3. So we have:
(2/53)π = (1/3)πr^2h
We also know that the height of the cone is equal to the diameter of the sphere, which means:
h = 2r
Substituting this into the volume formula, we have:
(2/53)π = (1/3)πr^2(2r)
2/53 = (2/3)r^3
r^3 = (2/53)(3/2)
r^3 = 6/159
r^3 = 2/53
r = (2/53)^(1/3)
Now that we know the radius of the sphere is (2/53)^(1/3) cm, we can find its volume using the formula for the volume of a sphere:
Vsphere = (4/3)πr^3
Substituting the value of r, we have:
Vsphere = (4/3)π((2/53)^(1/3))^3
Simplifying, we get the final answer for the volume of the sphere.
Whats the answer
I apologize for the confusion in my previous response. Let's calculate the answer.
We are given that the volume of the cone is (2/53)π cm^3. Therefore, we have:
Vcone = (2/53)π cm^3
Since the radius of the cone is the same as the sphere's radius, let's denote it as r.
The height of the cone is equal to the sphere's diameter, so we have:
h = 2r
The formula for the volume of a cone is:
Vcone = (1/3)πr^2h
Substituting the value of h, we get:
(2/53)π = (1/3)πr^2(2r)
Simplifying, we have:
(2/53)π = (4/3)πr^3
To find the volume of the sphere, we can set the volume of the cone equal to the volume of the sphere:
(2/53)π = (4/3)πr^3
Cancelling out the common factor of π, we have:
2/53 = (4/3)r^3
Solving for r, we find:
r^3 = (2/53) * (3/4)
r^3 = 6/212
r = (6/212)^(1/3)
Now that we know the radius of the sphere, we can calculate its volume using the formula for the volume of a sphere:
Vsphere = (4/3)πr^3
Substituting the value of r, we get:
Vsphere = (4/3)π * ((6/212)^(1/3))^3
After evaluating this expression, we can find the volume of the sphere.