Multiple choice

Curved surface area and total surface area of a cone are in ratio 13 : 18. What is the ratio of height to radius of the cone?

  1. 6 : 13

  2. 10 : 13

  3. 12 : 5

  4. 13 : 5

  5. 15 : 13
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C Correct answer
Explanation

Curved surface area is pi*r*l and total surface area is pi*r(l+r). The ratio pi*r*l / pi*r(l+r) = l / (l+r) = 13/18. This implies 18l = 13l + 13r, so 5l = 13r, or l/r = 13/5. Using l^2 = h^2 + r^2, we find h^2 = l^2 - r^2 = (13r)^2/25 - r^2 = 144r^2/25, so h/r = 12/5.

AI explanation

The total surface area of a cone is pi r l plus pi r squared, and the curved surface area is pi r l. Their ratio of 13 to 18 implies the base area to curved surface area ratio is 5 to 13, meaning pi r squared divided by pi r l equals 5 divided by 13. This gives the slant height l as 13 units. Using the Pythagorean theorem, the height squared equals 13 squared minus 5 squared, so the height is 12. The ratio of height to radius is 12 to 5.

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