SOLUTION: A solid body consists of a cylinder surmounted by a hemisphere of the same radius. The total length of the body, measured along the central axis of the cylinder is 10 cm. If the ra

Question 1139361: A solid body consists of a cylinder surmounted by a hemisphere of the same radius. The total length of the body, measured along the central axis of the cylinder is 10 cm. If the radius of the hemisphere is x cm show that the volume, v cm³, of the solid body is given by v = π(x²/3)(30-x).
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the volume of a cylinder is equal to pi * r^2 * h

r is the radius.
h is the height.

the volume of a sphere is equal to 4/3 * pi * r^3

the volume of a hemisphere is half that.

this makes the volume of the hemisphere equal to 4/6 * pi * r^3

this simplifies to v = 2/3 * pi * r^3

so far, you have:

the volume a cylinder is equal to pi * r^2 * h

the volume of a hemisphere is equal to 2/3 * pi * r^3

when the radius of the hemisphere is the same as the radius of the cylinder, and the radius of the hemisphere is equal to x, then the formula for both the hemisphere and the cylinder become:

v = pi * x^2 * h for the cylinder.

v = 2/3 * pi * x^3 for the hemisphere.

you are given that the hemisphere sits on top of the cylinder and that the overall height of both, measured from the center line of the cylinder, is equal to 10.

this means that part of the height is the radius of the hemisphere.

since the radius of the hemispher is x, then the overall height is composed of x plus (10 - x), with x being the part of the height that is the radius of the hemisphere and (10 - x) being the part of the height that is the height of the cylinder itself.

this means that the height of the cylinder is equal to 10 - x.

since h = (10 - x), then the formula for the volume of the cylinder becomes:

v = pi * r^2 * (10 - x).

the formula for the volume of the hemisphere remains the same at v = 2/3 * pi * x^3.

the overall volume for both is the volume of the cylinder plus the volume of the hemisphere.

this becomes:

v = pi * x^2 * (10 - x) + 2/3 * pi * x^3

factor out the pi to get v = pi * (x^2 * (10 - x) + 2/3 * x^3)

simplify this to get v = pi * (10x^2 - x^3 + 2/3 * x^3)

combine like terms to get v = pi ( 10x^2 - 1/3 * x^3)

factor out an x^2 to get v = pi * x^2 * (10 - 1/3 * x)

factor out a 1/3 to get v = pi * x^2 / 3 * (30 - x).

place parentheses around pi * x^2 to get v = pi * (x^2/3) * (30 - x).

this can also be shown as v = pi(x^2/3)(10-x).

that's your solution.

it shows that If the radius of the hemisphere is x cm. then the volume of the solid body, that is comprised of the cylinder plus the hemisphere on top of it, is given by v = π(x²/3)(30-x).
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