Multiple choice

The angle of elevation of the top of the tower is ${ 45 }^{ o }$ on walking up a slope inclined at an angle of ${ 30 }^{ o }$ to the horizontal a distance $20mt$, the angle of elevation of top of tower is observed to be ${ 60 }^{ o }$. Then the height of the tower

  1. $10\left( \sqrt { 3 } +1 \right) mt$
  2. $20\left( \sqrt { 3 } +1 \right) mt$
  3. $100\sqrt { 3 } mt$
  4. $50\left( 3+\sqrt { 3 } \right) $
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A Correct answer
Explanation

Let h be the height of the tower. Using trigonometry, the distance from the base is h/tan(45) = h. After walking 20m up a 30-degree slope, the new position is (20cos30, 20sin30) = (10sqrt3, 10). The new angle of elevation is 60 degrees, so tan(60) = (h - 10) / (h - 10sqrt3). sqrt(3) = (h - 10) / (h - 10sqrt3). h*sqrt(3) - 30 = h - 10. h(sqrt(3)-1) = 20. h = 20 / (sqrt(3)-1) = 20(sqrt(3)+1) / 2 = 10(sqrt(3)+1).

AI explanation

Let the tower's height be H and its base be at distance D from the starting point, placing the observer after walking 20 m at a horizontal distance of D + 10 sqrt(3) and a vertical height of 10. Initially, tan 45 degrees equals H divided by D, meaning H equals D. Applying the tangent ratio at the second position gives tan 60 degrees = sqrt(3), which equals (H - 10) divided by (D + 10 sqrt(3)). Substituting D with H and solving the resulting equation yields H = 10(sqrt(3) + 1) m.

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