Find the ratio of the volume of the cylinder to the volume of the hemisphere, given that the height,
h, of the cylinder is equal to the diameter, d, of the hemisphere
volume of cyl: pi r^2 h
but if h = d then h = 2r, so
pi * r^2 * 2r
= 2pi * r^3
Vol of hemisphere = (1/2) volume of sphere:
= 2/3 pi r^3
So the ratio is obvious from those results.
To find the ratio of the volume of the cylinder to the volume of the hemisphere, we first need to determine the formulas for calculating the volumes of both shapes.
The volume of a cylinder is given by V_cylinder = πr^2h, where r is the radius of the cylinder and h is the height (in this case, h = d, the diameter of the hemisphere).
The volume of a hemisphere is given by V_hemisphere = (2/3)πr^3, where r is the radius of the hemisphere.
Given that the height of the cylinder is equal to the diameter of the hemisphere (h = d), we can determine that the radius of the cylinder is half the diameter (r = d/2).
Substituting these values into the volume formulas, we get:
V_cylinder = π(d/2)^2 * d
= π(d^2/4) * d
= (π/4) * d^3
V_hemisphere = (2/3)π(d/2)^3
= (2/3)π(d^3/8)
= (π/12) * d^3
Now, we can find the ratio of the volume of the cylinder (V_cylinder) to the volume of the hemisphere (V_hemisphere) by dividing V_cylinder by V_hemisphere:
Ratio = V_cylinder / V_hemisphere
= [(π/4) * d^3] / [(π/12) * d^3]
= (π/4) * d^3 * (12/π) * (1/d^3)
= (3/1) * 1
= 3
Therefore, the ratio of the volume of the cylinder to the volume of the hemisphere is 3.