Multiple choice

A right circular cylinder has a height of 15 units and a radius of 7 units. A rectangular solid with a height of 12 units and a square base is placed in the cylinder such that each of the corners of the solid is tangential to the cylinder wall. Liquid is then poured into the cylinder such that it reaches the rim. The volume of the liquid is

  1. 147(5 π - 8) unit3

  2. 180( π - 5) unit3

  3. 49( 5π - 24) unit3

  4. 49( 15π - 8) unit3
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A Correct answer
Explanation

Volume of cylinder = pi * r^2 * h = pi * 49 * 15 = 735pi. The rectangular solid has a square base. If corners are tangential to the cylinder, the diagonal of the square base = diameter of cylinder = 14. Side of square = 14 / sqrt(2) = 7 * sqrt(2). Area of base = 98. Volume of solid = 98 * 12 = 1176. Liquid volume = 735pi - 1176 = 147(5pi - 8).

AI explanation

The total volume of the right circular cylinder is pi times the radius squared times the height, which equals pi times 7 squared times 15, simplifying to 735 pi. The square base of the rectangular solid fits inside the cylinder, so its diagonal equals the diameter of the cylinder (14 units), meaning the side of the square is 14 divided by the square root of 2, and the base area is 98 square units. The volume of the rectangular solid is 98 times 12, which equals 1176. The volume of the liquid is the total cylinder volume minus the solid volume, which equals 735 pi minus 1176; factoring out 147 gives 147 times (5 pi minus 8) unit3.

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