The volume of a sphere is 48 m3. What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?
I don't agree with the bot
4/3 π r^3 = 48
r^3 = 11.459
r = 2.2545 <--- radius of sphere
V of cyl = π r^2 h
= π(2.2545^2)(4.509) = 71.9999 or 72 m^3
What is the explanation
To find the volume of the cylinder, we need to use the formula for the volume of a cylinder, which is V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the radius of the sphere is the same as the radius of the cylinder, we can say that r_sphere = r_cylinder.
We're also given that the height of the cylinder is equal to the diameter of the sphere. Remember that the diameter is twice the radius. So, h_cylinder = 2r_sphere.
Now, let's substitute these values into the formula for the volume of a cylinder:
V_cylinder = πr_cylinder^2 * h_cylinder
= π(r_sphere)^2 * (2r_sphere)
= 2πr_sphere^3
However, we don't have the value of the radius of the sphere. To find it, we can use the formula for the volume of a sphere, which is V_sphere = (4/3)πr_sphere^3.
Given that the volume of the sphere is 48 m^3, we can rearrange the formula and solve for the radius:
48 = (4/3)πr_sphere^3
Multiply both sides by 3/4π:
r_sphere^3 = (3/4π) * 48
Divide both sides by (3/4π) to find r_sphere cubed:
r_sphere^3 = 64
To isolate r_sphere, take the cube root of both sides:
r_sphere = ∛64
= 4
Now that we have the value of r_sphere, we can substitute it into the formula for the volume of the cylinder:
V_cylinder = 2πr_sphere^3
= 2π(4)^2 * (2*4)
= 32π
Therefore, the volume of the cylinder is 32π cubic units, where π is approximately 3.14.