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?
The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius and h is the height. Given that the volume of the cylinder is 36 cm^3, we can set up the equation as follows:
36 = πr^2h
Since the height of the cylinder is equal to the sphere's diameter, we can write h = 2r, where r is the radius of the sphere.
Substituting this into the equation, we have:
36 = πr^2(2r)
Multiplying the terms inside the parentheses, we get:
36 = 2πr^3
Now, let's solve for the volume of the sphere (Vsphere) using the formula Vsphere = (4/3)πr^3.
Dividing both sides of the equation by 2π, we get:
18 = r^3
Taking the cube root of both sides, we get:
r = 3
Substituting this value into the formula for the volume of the sphere, we have:
Vsphere = (4/3)π(3)^3 = (4/3)π(27) = 36π cm^3
Therefore, the volume of the sphere is 36π cm^3.
Wrong
Apologies for the mistake. Let's approach the problem again.
The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius and h is the height. Given that the volume of the cylinder is 36 cm^3, we can set up the equation as follows:
36 = πr^2h
Since the height of the cylinder is equal to the sphere's diameter, we can write h = 2r, where r is the radius of the sphere.
Substituting this into the equation, we have:
36 = πr^2(2r)
Multiplying the terms inside the parentheses, we get:
36 = 2πr^3
Now, let's solve for the radius of the sphere (rsphere) using the equation above:
2πr^3 = 36
Divide both sides by 2π:
r^3 = 18/π
Take the cube root of both sides:
r = (18/π)^(1/3)
Now, let's find the volume of the sphere (Vsphere) using the formula Vsphere = (4/3)πr^3:
Vsphere = (4/3)π[(18/π)^(1/3)]^3
Simplifying the equation:
Vsphere = (4/3)π(18/π)
The π cancels out, and we are left with:
Vsphere = (4/3)x 18
Vsphere = 24 cm^3
Therefore, the volume of the sphere with the same radius as the cylinder and a height equal to the sphere's diameter is 24 cm^3.