Multiple choice

If the ratio of curved surface areas of the frustums of two cones is 2 : 3 and the ratio of their slant heights is 8 : 3, then the ratio of sum of the radii of the frustums of the 1st cone to that of the 2nd cone is

  1. 2 : 1

  2. 1 : 2

  3. 4 : 1

  4. 1 : 4

  5. None of these
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D Correct answer
Explanation

The curved surface area of a frustum is pi * (r1 + r2) * l. Given the ratio of areas is 2:3 and slant heights is 8:3, we have (r1a + r2a) * 8 / ((r1b + r2b) * 3) = 2/3. Solving for the ratio (r1a + r2a) / (r1b + r2b) gives 2/3 * 3/8 = 6/24 = 1/4.

AI explanation

The formula for the curved surface area of a frustum is pi times the sum of the radii times the slant height. Let the sums of the radii be R1 and R2, and the slant heights be L1 and L2, which gives the equation R1 times L1 divided by R2 times L2 equals 2 divided by 3. Substituting the given slant height ratio of 8 to 3, we have R1 divided by R2 times 8 divided by 3 equals 2 divided by 3. Solving this yields R1 divided by R2 equals 1 divided by 4, making the ratio 1 to 4.

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