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.
Vol. of cylinder = ¡Ç(r^2)h and vol. of hemi-sphere=(¡Çh^3)/12 and hence their ratio = 12(r/h)^2 where r, h are the radius & and the height of the cylinder
To find the ratio of the volume of the cylinder to the volume of the hemisphere, we need to calculate the volumes of each shape and then divide the volume of the cylinder by the volume of the hemisphere.
First, let's calculate the volume of the cylinder. The volume of a cylinder is given by the formula V = πr^2h, where r is the radius of the base and h is the height of the cylinder. Since the height of the cylinder is equal to the diameter of the hemisphere, we can say that h = d.
The radius of the cylinder's base is half the diameter, so the radius, r, can be expressed as r = d/2.
Substituting the values of r and h into the volume formula, we get:
Vcylinder = π(d/2)^2h = πd^2h/4.
Next, let's calculate the volume of the hemisphere. The volume of a hemisphere is given by the formula V = (2/3)πr^3. Since the radius of the hemisphere is half the diameter, we can say that r = d/2.
Substituting the value of r into the volume formula, we get:
Vhemisphere = (2/3)π(d/2)^3 = (2/3)πd^3/8.
Now, we can find the ratio of the volume of the cylinder to the volume of the hemisphere by dividing the volume of the cylinder by the volume of the hemisphere:
(Vcylinder/Vhemisphere) = [πd^2h/4] / [(2/3)πd^3/8].
To simplify this expression, we can cancel out common factors:
(Vcylinder/Vhemisphere) = (3d/4) * (8/2d^3) = 3/2d^2.
Therefore, the ratio of the volume of the cylinder to the volume of the hemisphere is 3/2d^2.