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.
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
3 answers
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.
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.