Multiple choice

The height of a tower is half the height of the flagstaff at its top. The angle of elevation of the top of the tower as seen from a distance of $10$ metres from its foot is ${30}^{o}$. Find the angle of elevation of the top of the flagstaff from the same point.

  1. Angle of elevation is ${35}^{o}$
  2. Angle of elevation is ${52}^{o}$
  3. Angle of elevation is ${55}^{o}$
  4. Angle of elevation is ${60}^{o}$
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D Correct answer
Explanation

Let tower height be h, flagstaff height be 2h. From 10m away, tan(30) = h/10, so h = 10 * tan(30) = 10 / sqrt(3). The flagstaff is on top, so total height is 3h. Tan(theta) = 3h / 10 = 3 * (10 / sqrt(3)) / 10 = 3 / sqrt(3) = sqrt(3). Theta = arctan(sqrt(3)) = 60 degrees.

AI explanation

Using the angle of elevation of 30 degrees from 10 metres away, the height of the tower is found using the tangent ratio: tan 30 degrees equals height divided by 10, giving a tower height of 10 divided by root 3 metres. Since the tower is half the height of the flagstaff, the flagstaff is twice as tall, making its height 20 divided by root 3 metres. The top of the flagstaff is therefore at a total height of 10 divided by root 3 plus 20 divided by root 3, which equals 30 divided by root 3 or 10 root 3 metres. Let the new angle of elevation be theta; then tan theta equals 10 root 3 divided by 10, which equals root 3. The angle whose tangent is root 3 is 60 degrees, so the angle of elevation is 60 degrees.

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