Multiple choice

A right angle triangle with perpendicular and base as $8$ and $6$ $cm$ respectively, is rotated about its longest side. Find the volume of the solid generated.

  1. $50.28\pi$
  2. $150.85\pi$
  3. $76.8\pi$

  4. None of these
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C Correct answer
Explanation

Rotating a right triangle about its hypotenuse creates two cones sharing a base. Hypotenuse = sqrt(8^2 + 6^2) = 10. The altitude to the hypotenuse is h = (8 * 6) / 10 = 4.8. The radius of the common base is 4.8. Volume = (1/3) * pi * r^2 * h1 + (1/3) * pi * r^2 * h2 = (1/3) * pi * r^2 * (h1 + h2) = (1/3) * pi * (4.8)^2 * 10 = 76.8 * pi.

AI explanation

For the given right triangle with perpendicular 8 cm and base 6 cm, the hypotenuse is calculated using the Pythagorean theorem as 10 cm. Rotating the triangle about its hypotenuse forms a double cone system, where the radius of the common base is the altitude to the hypotenuse, calculated as (8 * 6) / 10 = 4.8 cm. This altitude splits the hypotenuse into two segments of 6.4 cm and 3.6 cm, serving as the heights of the two cones. The total volume is the sum of the volumes of these two cones, calculated as (1/3) * pi * 4.8^2 * 6.4 + (1/3) * pi * 4.8^2 * 3.6, which simplifies to (1/3) * pi * 4.8^2 * 10, equaling 76.8*pi.

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