Multiple choice

Two observers are stationed due north of a tower at a distance of 20 m form each other. if the elevations of the tower observed by them are $\displaystyle 30^{\circ}$ and $\displaystyle 45^{\circ}$ respectively, then the height of the tower is

  1. $10$ m$
  2. $16.32$ m
  3. $\displaystyle 10(\sqrt{3}+1)$ m
  4. $30$ m
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C Correct answer
Explanation

Let the nearer observer be x m from the tower. The 45-degree elevation gives height h = x. The other observer is x+20 m away, so tan(30 degrees) = h/(h+20). Solving gives h = 10(sqrt(3)+1) m.

AI explanation

Let the tower height be h and the closer observer be 20 m from the farther one, where the distance from the tower base to the closer observer is d. Using the elevation angle of 45 degrees at the closer observer, we get tan(45) = h divided by d, meaning h = d. At the farther observer, the distance is d + 20, so tan(30) = h divided by (d + 20). Substituting h for d yields 1 divided by sqrt(3) = h divided by (h + 20), which solves to h = 10(sqrt(3) + 1) m.

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