Multiple choice

The surface area of the frustum cone is $297 cm^{2}$ whose top circle radius is $2$ cm and bottom circle radius is $5$ cm. Find its slant height. (Use $\pi = 3$)

  1. $10$ m
  2. $8$ cm
  3. $9$ cm
  4. $10$ cm
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D Correct answer
Explanation

The surface area of a frustum (excluding bases) is pi * l * (r1 + r2). Given SA = 297, pi = 3, r1 = 2, r2 = 5, we have 297 = 3 * l * (2 + 5). Thus, 297 = 21 * l, so l = 297 / 21 = 14.14. However, checking the total surface area formula (including bases): SA = pi * l * (r1 + r2) + pi * r1^2 + pi * r2^2. 297 = 3 * l * 7 + 3 * 4 + 3 * 25 = 21l + 12 + 75 = 21l + 87. 210 = 21l, so l = 10.

AI explanation

The total surface area of a frustum of a cone equals pi times the sum of the radii times the slant height plus the areas of the two circles. Using the given surface area of 297, we set up the equation 297 equals 3 times (2 plus 5) times the slant height plus 3 times 2 squared plus 3 times 5 squared. This simplifies to 297 equals 21 times the slant height plus 12 plus 75, leaving 297 equal to 21 times the slant height plus 87. Subtracting 87 from 297 gives 210, and dividing by 21 results in a slant height of 10 cm.

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