Multiple choice

The angle of elevation of an electric pole from a point to the ground is $60^{ o}$ and from a point $B$ towards the pole on the line joining the foot of the pole to the point is $75^{o}$. If the distance $AB=a$, then the height of the pole is:

  1. $\displaystyle \frac{a(3+2\sqrt{3})}{2}$
  2. $a(4+2\sqrt{3})$
  3. $\displaystyle \frac{a(2+\sqrt{3})}{2}$
  4. $\displaystyle \frac{a(2\sqrt{3}-3)}{2}$
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A Correct answer
Explanation

Let height be h. Distance from pole = h/tan(60) = h/sqrt(3). Distance from B = h/tan(75). a = h/tan(60) - h/tan(75) = h(1/sqrt(3) - (2-sqrt(3))). Solving for h gives a(3+2sqrt(3))/2.

AI explanation

Let the height of the pole be h and the distance from the foot of the pole to point B be d. From the right triangles formed at points A and B, we have the equations tan 60 degrees equals h divided by (d plus a) and tan 75 degrees equals h divided by d. Using the exact values for these tangents, the square root of 3 equals h divided by (d plus a) and 2 plus the square root of 3 equals h divided by d, which allows us to express d as h divided by (2 plus the square root of 3). Substituting this expression for d into the first equation and solving for h gives h equals a multiplied by (3 plus 2 times the square root of 3) divided by 2.

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