Multiple choice

The radius and height of a right circular cone are in the ratio $3:4$. If its volume is $96\pi cm^3$, what is its slant height?

  1. $8$cm
  2. $9$cm
  3. $10$cm
  4. $12$cm
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C Correct answer
Explanation

Let radius r = 3k and height h = 4k. Volume = (1/3) * pi * r^2 * h = (1/3) * pi * (9k^2) * (4k) = 12 * pi * k^3. Given 12 * pi * k^3 = 96 * pi, so k^3 = 8 and k = 2. Thus, r = 6 and h = 8. The slant height l = sqrt(r^2 + h^2) = sqrt(36 + 64) = 10 cm.

AI explanation

Let the radius be 3x and the height be 4x based on the given ratio. Using the cone volume formula V equals (1/3)πr^2h, we substitute the values to get 96π equals (1/3)π times (3x)^2 times 4x. Solving this yields 12x^3 equals 96, so x^3 equals 8 and x equals 2. This makes the radius 6 cm and the height 8 cm, so the slant height l equals the square root of (6^2 plus 8^2), which equals 10 cm.

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