Multiple choice

From a point on a horizontal plane, the elevation of the top of a hill is $45^o$. The elevation becomes $75^o$ after walking a distance of 500 m up a slope of inclined at an angle of $15^o$ to the horizon. The height of the hill is

  1. $500 \sqrt{6}$
  2. $500 \sqrt{3}$
  3. $250 \sqrt{6}$
  4. $250 \sqrt{3}$
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C Correct answer
Explanation

Using trigonometry, the height h = 500 * sin(15) + (500 * cos(15) * tan(75)) / (tan(75) - tan(45)). This simplifies to 250 * sqrt(6).

AI explanation

Starting at point A, the elevation is 45 degrees. After walking 500 m up a 15 degree incline to point B, the elevation becomes 75 degrees. In triangle AOB, angle O is 90 degrees, angle A is 30 degrees, and angle B is 60 degrees. Using the sine rule, OA divided by sin 60 degrees equals 500 divided by sin 30 degrees, making OA equal to 500 times the square root of 3. The height of the hill is found by adding the vertical component OA to the vertical height gained at B, which results in 250 times the square root of 6.

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