Multiple choice

A right circular cone of vertical height $24 cm$ has volume $\displaystyle 1232cm^{3}$. Its curved surface area is

  1. $\displaystyle 450cm^{2}$
  2. $\displaystyle 500cm^{2}$
  3. $\displaystyle 550cm^{2}$
  4. $\displaystyle 600cm^{2}$
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C Correct answer
Explanation

Volume V = (1/3) * pi * r^2 * h = 1232. (1/3) * (22/7) * r^2 * 24 = 1232. 8 * (22/7) * r^2 = 1232. r^2 = 1232 * 7 / 176 = 49. So r = 7. Slant height l = sqrt(r^2 + h^2) = sqrt(49 + 576) = sqrt(625) = 25. Curved surface area = pi * r * l = (22/7) * 7 * 25 = 550.

AI explanation

Using the cone volume formula V equals one third times pi times the square of the radius times the height, we set 1232 equal to one third times 22 over 7 times r squared times 24. Solving this equation gives r squared equal to 49, meaning the radius is 7 cm. The slant height l is found using the Pythagorean theorem as the square root of 24 squared plus 7 squared, which equals 25 cm. The curved surface area is pi times r times l, becoming 22 over 7 times 7 times 25, which equals 550 square centimeters.

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