Multiple choice

The radius and height of a right circular cone are in th ratio $2:3$. Find the slant height if its volume is $100.48\ cm^3$. (Take $\pi =3.14$).

  1. $2\sqrt{13}cm$
  2. $4\sqrt{13}cm$
  3. $2\sqrt{3}cm$

  4. None of these
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A Correct answer
Explanation

Radius r = 2k, height h = 3k. Volume = (1/3) * pi * r^2 * h = (1/3) * 3.14 * (4k^2) * 3k = 12.56 * k^3 = 100.48. k^3 = 8, so k = 2. r = 4, h = 6. Slant height l = sqrt(r^2 + h^2) = sqrt(16 + 36) = sqrt(52) = 2 * sqrt(13).

AI explanation

With the radius and height in a 2:3 ratio, we let r = 2x and h = 3x. Using the cone volume formula V = (1/3) * pi * r^2 * h, we substitute the values to get 100.48 = (1/3) * 3.14 * (2x)^2 * (3x), which simplifies to 100.48 = 12.56 * x^3, giving x = 2. Therefore, the radius is 4 cm and the height is 6 cm. Applying the slant height formula l = sqrt(r^2 + h^2), we find l = sqrt(16 + 36) = sqrt(52), which simplifies to 2 * sqrt(13) cm.

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