The volume of a cone is 253 π cm3. 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) Responses 256 π cm3 Start Fraction 25 over 6 End Fraction pi cm cubed 25 π cm3 25 pi cm cubed 252 π cm3 Start Fraction 25 over 2 End Fraction pi cm cubed 503 π cm3
The volume of the cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. We are given that the volume of the cone is 253π cm^3.
253π = (1/3)πr^2h
To find the volume of the sphere, we need to find the radius and height of the cone.
Since the height of the cone is equal to the diameter of the sphere, we can find the radius of the cone using the formula:
d = 2r
Substituting h = d into the formula for the volume of the cone:
253π = (1/3)πr^2d
253 = (1/3)r^2d
Since d = 2r:
253 = (1/3)r^2(2r)
253 = (2/3)r^3
r^3 = (3/2)253
r^3 = 379.5
r ≈ 7.17
Now we can find the volume of the sphere using the formula V = (4/3)πr^3:
V_sphere = (4/3)π(7.17)^3
V_sphere ≈ 4(22/7)(7.17)^3
V_sphere ≈ 4(22)(7.17)^2(7.17)
V_sphere ≈ 254.54π
Therefore, the volume of the sphere is approximately 254.54π cm^3.
wrong
Apologies for the incorrect response.
Let's solve this problem step by step:
The formula for the volume of a cone is V_cone = (1/3)πr^2h, and we are given that the volume of the cone is 253π cm^3.
253π = (1/3)πr^2h
Since the height of the cone is equal to the diameter of the sphere, we can say h = 2r.
253π = (1/3)πr^2(2r)
253π = (2/3)πr^3
To find the radius, we can rearrange the equation:
r^3 = (3/2) * 253
r^3 = 379.5
Taking the cube root of both sides, we find:
r ≈ 7.54
Now, let's find the volume of the sphere using the formula V_sphere = (4/3)πr^3:
V_sphere = (4/3)π(7.54)^3
V_sphere ≈ 4(22/7)(7.54)^2(7.54)
V_sphere ≈ 355.33π
Therefore, the volume of the sphere is approximately 355.33π cm^3.
To find the volume of the sphere, we need to use the given information about the cone.
Let's start by finding the radius of the cone. The formula for the volume of a cone is:
V_cone = (1/3) * π * r^2 * h,
where V_cone is the volume of the cone, π is pi, r is the radius, and h is the height of the cone.
Given that the volume of the cone is 253π cm^3, we can set up the equation:
253π = (1/3) * π * r^2 * h.
Since we are told that the height of the cone is equal to the sphere's diameter, we can express the height as h = 2r, where r is the radius.
Now our equation becomes:
253π = (1/3) * π * r^2 * (2r).
Simplifying further:
253π = (2/3) * π * r^3.
Cancelling out π, we have:
253 = (2/3) * r^3.
To find the radius, we can rearrange the equation:
r^3 = (3/2) * 253.
r^3 = 759/2.
Taking the cube root of both sides:
r ≈ 8.62 cm.
Now that we know the radius of the cone, we can find the volume of the sphere. The formula for the volume of a sphere is:
V_sphere = (4/3) * π * r^3,
where V_sphere is the volume of the sphere and r is the radius.
Plugging in the value we found for r:
V_sphere = (4/3) * π * (8.62)^3.
Evaluating this expression, we get approximately:
V_sphere ≈ 503.29π cm^3.
Therefore, the volume of the sphere is approximately 503π cm^3.