The volume of a sphere is 72 m3 . What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius of the sphere.
In this case, if the volume of the sphere is 72 m^3, we have:
72 m^3 = (4/3)πr^3
To find the volume of the cone, we need to first find the radius of the sphere using the given volume.
Let's solve for r:
72 m^3 = (4/3)πr^3
Multiplying both sides of the equation by 3/4 to isolate r, we get:
r^3 = (3/4)*(72 m^3) = 54 m^3
Taking the cube root of both sides to solve for r, we have:
r = ∛(54 m^3) ≈ 3.78 m
Now that we know the radius of the sphere is approximately 3.78 m, we can find the volume of the cone.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius of the base and h is the height.
In this case, the radius of the cone is the same as the sphere's radius (approximately 3.78 m), and the height of the cone is equal to the sphere's diameter, which is twice the radius of the sphere:
h = 2 * r ≈ 2 * 3.78 m ≈ 7.57 m
Now let's substitute the values into the cone volume formula:
V_cone = (1/3)πr^2h
V_cone = (1/3)π(3.78 m)^2 * 7.57 m
V_cone ≈ (1/3)π(14.29 m^2) * 7.57 m
V_cone ≈ 150.36 m^3
Therefore, the volume of the cone is approximately 150.36 m^3.