Question

A cone is placed inside a cylinder. The cone has half the radius of the cylinder, but the height of each figure is the same. The cone is tilted at an angle so its peak touches the edge of the cylinder’s base. What is the volume of the space remaining in the cylinder after the cone is placed inside it? (Please explain and show work)

Answer 1

Answer: (47/48)*pi*r^2*h

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Work Shown:

A = volume of the cylinder

A = pi*(radius)^2*(height)

A = pi*r^2*h

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The cone's base has diameter r/2, so its radius is (1/2)*(r/2) = r/4

B = volume of the cone

B = (1/3)*(volume of cylinder with radius r/4 and height h)

B = (1/3)pi*(r/4)^2*h

B = (1/3)*pi*(r^2)/(4^2)*h

B = (1/3)*pi*(r^2)/16*h

B = (1/3)*(1/16)*pi*r^2*h

B = (1/48)pi*r^2*h

B = (1/48)*A

B = (1/48)*(volume of cylinder)

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C = volume of the empty space/air inside cylinder, but outside cone

C = A-B

C = A - (1/48)*A

C = (48/48)A - (1/48)A

C = (48A)/48 - (A/48)

C = (48A-A)/48

C = 47A/48

C = (47/48)*A

C = (47/48)*pi*r^2*h