Multiple choice

The radius and the height of a right circular cone are in the ratio of 3 : 5 If its volume is 120$\displaystyle \pi $ cu m its slant height is

  1. $\displaystyle 3\sqrt{34}m $
  2. $\displaystyle 2\sqrt{28}m $
  3. $\displaystyle 2\sqrt{44}m $
  4. $\displaystyle 2\sqrt{34}m $
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D Correct answer
Explanation

Ratio r:h = 3:5, so r = 3x, h = 5x. Volume = (1/3) * pi * r^2 * h = 120 * pi. (1/3) * pi * (9x^2) * (5x) = 120 * pi. 15x^3 = 120, so x^3 = 8, x = 2. Thus r = 6, h = 10. Slant height l = sqrt(r^2 + h^2) = sqrt(36 + 100) = sqrt(136) = 2 * sqrt(34).

AI explanation

Let the radius be 3x and the height be 5x. Using the cone volume formula V = (1/3) * pi * r^2 * h, we get (1/3) * pi * (3x)^2 * (5x) = 120 * pi. Solving 15x^3 = 120 yields x = 2, meaning the radius is 6 m and the height is 10 m. Applying the Pythagorean theorem for the slant height gives l = sqrt(6^2 + 10^2), which equals sqrt(136) or 2 * sqrt(34) m.

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