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 cm3 . 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) Responses 18 cm3 18 cm cubed 24 cm3 24 cm cubed 54 cm3 54 cm cubed 12 cm3
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
In this problem, we are given that the volume of the cylinder is 36 cm^3 and the height of the cylinder is equal to the sphere's diameter.
Let's assume that the radius of the cylinder is r cm. Since the height of the cylinder is equal to the sphere's diameter, it means the height of the cylinder is 2r cm.
We can substitute these values into the volume formula to find the value of r:
36 = π(r^2)(2r)
36 = 2πr^3
Divide both sides by 2π:
18 = r^3
Take the cube root of both sides:
r = 3
Therefore, the radius of the cylinder is 3 cm.
Now, we need to find the volume of the sphere using the same radius. The formula for the volume of a sphere is V = (4/3)πr^3.
Substituting the value of r, we have:
V = (4/3)π(3^3)
V = (4/3)π(27)
V = (4/3) * 27π
V = 36π
Therefore, the volume of the sphere is 36π cm^3.
Since the options are given in cubic centimeters, we can express π as its approximate value, 3.14:
V = 36 * 3.14
V ≈ 113.04 cm^3
Therefore, the volume of the sphere is approximately 113.04 cm^3.
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I apologize for the mistake. Let's solve the problem correctly.
We are given that the volume of the cylinder is 36 cm^3 and the height of the cylinder is equal to the sphere's diameter.
The formula for the volume of a cylinder is V_cylinder = πr^2h, where r is the radius and h is the height.
We can rewrite the volume of the cylinder as:
36 = πr^2(2r)
Now, let's solve for r:
36 = 2πr^3
Divide both sides by 2π:
18 = r^3
Take the cube root of both sides:
r = 3
Therefore, the radius of the cylinder is 3 cm.
Now, let's find the volume of the sphere using the same radius. The formula for the volume of a sphere is V_sphere = (4/3)πr^3.
Substituting the value of r, we have:
V_sphere = (4/3)π(3^3)
V_sphere = (4/3)π(27)
V_sphere = (4/3) * 27π
V_sphere = 36π
Therefore, the volume of the sphere is 36π cm^3.
Since the options are given in cubic centimeters, we can evaluate the value of 36π:
V_sphere ≈ 113.097 cm^3
Therefore, the volume of the sphere is approximately 113.097 cm^3.
To solve this problem, we need to use the relationship between the volumes of a cylinder and a sphere.
The formula for the volume of a cylinder is V_cylinder = π * r^2 * h, where r is the radius and h is the height of the cylinder.
The volume of the cylinder is given as 36 cm^3, which means V_cylinder = 36 cm^3. We are also given that the height of the cylinder is equal to the diameter of the sphere.
Now, let's consider the formula for the volume of a sphere, which is V_sphere = (4/3) * π * r^3, where r is the radius of the sphere.
Since the radius of the sphere is the same as the radius of the cylinder, we can set the radius of the cylinder as r.
Since the height of the cylinder is equal to the diameter of the sphere, we can say that h = 2r.
Substituting these values into the formula for the volume of the cylinder, we have:
36 cm^3 = π * r^2 * 2r
36 = π * 2r^3
Now, let's isolate r:
r^3 = 36 / (2π)
r^3 = 18 / π
r ≈ 2.707
Now, let's substitute this value of r into the formula for the volume of the sphere:
V_sphere = (4/3) * π * r^3
V_sphere ≈ (4/3) * π * (2.707)^3
V_sphere ≈ 4 * 3.14 * 20.083
V_sphere ≈ 251.33 cm^3
Therefore, the volume of the sphere, when the radius is the same as the cylinder's and the height of the cylinder is equal to the sphere's diameter, is approximately 251.33 cm^3.