Multiple choice

The angle elevation of the top of a tower from two points situated at distances 36 m and 64 m from its base and in the same straight line with it are complementary. What is the height of the tower?

  1. 50 m

  2. 48 m

  3. 25 m

  4. 24 m
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B Correct answer
Explanation

Let the height be h. The angles of elevation are complementary, so tan(theta) = h/36 and tan(90-theta) = h/64. Since tan(90-theta) = cot(theta) = 1/tan(theta), we have h/64 = 36/h, which leads to h^2 = 36 * 64 = 2304. Thus, h = sqrt(2304) = 48 m.

AI explanation

Let the height of the tower be H and the complementary angles of elevation be alpha and (90 degrees minus alpha). Using basic trigonometric ratios, we have tan(alpha) equals H divided by 36 and tan(90 degrees minus alpha) equals H divided by 64, which means cot(alpha) equals H divided by 64. Multiplying these two equations gives tan(alpha) multiplied by cot(alpha) equals H squared divided by the product of 36 and 64. Therefore, one equals H squared divided by 2304, so H equals 48 m.

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