Multiple choice

A right circular cone and a right circular cylinder have equal base and equal height. If the radius of the base and the height are in the ratio $5:12$ respectively, then the ratio of the total surface area of the cylinder to that of the cone is __________.

  1. $3:1$
  2. $13:9$
  3. $17:9$
  4. $34:9$
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C Correct answer
Explanation

For cylinder: r=5k, h=12k. TSA = 2*pi*r(r+h) = 2*pi*5k(17k) = 170*pi*k^2. For cone: r=5k, h=12k, slant l=13k. TSA = pi*r(r+l) = pi*5k(18k) = 90*pi*k^2. Ratio = 170/90 = 17:9.

AI explanation

Let the radius be 5x and the height be 12x, making the slant height of the cone sqrt((5x)^2 + (12x)^2) = 13x. The total surface area of the cylinder is 2 * pi * r * (h + r), which equals 2 * pi * 5x * (12x + 5x) = 170 * pi * x^2. The total surface area of the cone is pi * r * (l + r), which equals pi * 5x * (13x + 5x) = 90 * pi * x^2. The ratio of their total surface areas is 170 : 90, which simplifies to 17 : 9.

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