Multiple choice

Two poles standing on a horizontal ground are of heights $5m$ and $10m$ respectively. The line joining their tops makes an angle of ${15}^{o}$ with ground. Then the distance (in m) between the poles, is:

  1. $\cfrac { 5 }{ 2 } \left( 2+\sqrt { 3 } \right) $
  2. $5\left( \sqrt { 3 } +1 \right) $
  3. $5\left( 2+\sqrt { 3 } \right) $
  4. $10\left( \sqrt { 3 } -1 \right) $
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C Correct answer
Explanation

Let the poles be AB=5 and CD=10. The distance between them is x. The line joining the tops makes an angle of 15 degrees with the ground. This means the angle of elevation of the top of the taller pole from the top of the shorter pole is 15 degrees. tan(15) = (10-5)/x = 5/x. x = 5/tan(15). tan(15) = 2 - sqrt(3). x = 5 / (2 - sqrt(3)) = 5(2 + sqrt(3)).

AI explanation

The horizontal distance between the two poles is equal to the difference in their heights divided by the tangent of the elevation angle. The difference in heights is 10 m minus 5 m, which equals 5 m. The tangent of 15 degrees can be evaluated using the tangent difference formula as tan(45 degrees minus 30 degrees), which equals (1 minus 1/square root of 3) divided by (1 + 1/square root of 3), simplifying to 2 minus square root of 3. Dividing 5 m by (2 minus square root of 3) requires rationalization, resulting in 5 times (2 plus square root of 3) m.

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