Multiple choice

The angle of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at a distance 9 ft and 16 ft respectively are complementary angles. Then the height of the tower is:

  1. 9 ft

  2. 12 ft

  3. 16 ft

  4. 144 ft
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let height be h. tan(theta) = h/9, tan(90-theta) = h/16. cot(theta) = h/16. tan(theta) * cot(theta) = 1 = (h/9) * (h/16) = h^2 / 144. h^2 = 144, h = 12.

AI explanation

Let the height of the tower be H, and let the distances from the foot of the tower to the two points be 9 ft and 16 ft. Let the angles of elevation at these points be alpha and beta respectively. We are given that the angles are complementary, so alpha plus beta equals 90 degrees, which means beta equals 90 degrees minus alpha. Using the tangent ratio for both angles gives tan(alpha) = H / 9 and tan(beta) = H / 16. Substituting beta with 90 degrees minus alpha gives tan(90 degrees minus alpha) = H / 16. Using the co-function identity, tan(90 degrees minus alpha) equals cot(alpha), so we write cot(alpha) = H / 16. Since cot(alpha) is the reciprocal of tan(alpha), we have 9 / H = H / 16. Cross-multiplying this equation yields H squared equals 144. Taking the square root of both sides gives the height of the tower as 12 ft.

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