Multiple choice

A right circular cone of maximum size, was carved out from a right circular cylinder (keeping the height and the radius as same as of the cylinder). The difference between the volumes of the cylinder and the conical structure was 4928 cm3. The respective ratio between the height and radius is 6 : 7. What is the height of the cylinder?

  1. 6cm

  2. 18cm

  3. 12cm

  4. 25cm

  5. 34cm
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C Correct answer
Explanation

Volume of cylinder = pi * r^2 * h. Volume of cone = (1/3) * pi * r^2 * h. Difference = (2/3) * pi * r^2 * h = 4928. Given h/r = 6/7, let h=6k, r=7k. (2/3) * (22/7) * (49k^2) * (6k) = 4928. 44 * 7 * 2 * k^3 = 4928. 616 * k^3 = 4928. k^3 = 8, so k = 2. Height = 6 * 2 = 12.

AI explanation

The difference between the volumes of a cylinder and a cone with the same radius and height is two-thirds times pi times radius squared times height. Substituting the given difference of 4928 and the ratio of height to radius as 6 to 7, let the radius be 7r and the height be 6r. The equation becomes two-thirds times 22 over 7 times 49r squared times 6r equals 4928, simplifying to 616r cubed equals 4928, which means r cubed is 8 and r is 2. Since the height is 6r, the height is 6 times 2, giving 12 cm. The height of the cylinder is 12 cm.

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