Multiple choice

A right triangle $ABC$ with sides $5cm, 12cm$ and $13cm$ is revolved about the side $12cm$.If the triangle $ABC$ is revolved about the side $5cm$, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids obtained.

  1. $5:12$
  2. $12:5$
  3. $13:12$
  4. $5:11$
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A Correct answer
Explanation

When a right triangle is revolved about a leg, it forms a cone where the other leg is the radius and the revolved leg is the height. For the first case (revolved about 12cm), radius r1=5, height h1=12, volume V1 = (1/3) * pi * 5^2 * 12 = 100pi. For the second case (revolved about 5cm), radius r2=12, height h2=5, volume V2 = (1/3) * pi * 12^2 * 5 = 240pi. The ratio V1:V2 is 100:240, which simplifies to 5:12.

AI explanation

When the right triangle is revolved about its 12 cm side, it forms a cone with a radius of 5 cm and a height of 12 cm, yielding a volume of (1/3) * pi * 5^2 * 12. Revolving the triangle about the 5 cm side forms a cone with a radius of 12 cm and a height of 5 cm, yielding a volume of (1/3) * pi * 12^2 * 5. The ratio of these two volumes is ((1/3) * pi * 25 * 12) to ((1/3) * pi * 144 * 5), which simplifies to 5:12.

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