The shadow of a tower standing on a level ground is found to be 40 m longer when Suns altitude is 30 than when it was 60. Find the height of the tower.
Question
The shadow of a tower standing on a level ground is found to be 40 m longer when Suns altitude is 30 than when it was 60. Find the height of the tower.
Solution
To solve this problem, we will use the properties of right triangles and the trigonometric functions, specifically the tangent function which is the ratio of the opposite side to the adjacent side in a right triangle.
Step 1: Let's denote the height of the tower as h. When the Sun's altitude is 60 degrees, the length of the shadow is x. So, we can write the first equation from the tangent of 60 degrees:
tan(60) = h / x sqrt(3) = h / x => x = h / sqrt(3) ---- (equation 1)
Step 2: When the Sun's altitude is 30 degrees, the length of the shadow is x + 40. So, we can write the second equation from the tangent of 30 degrees:
tan(30) = h / (x + 40) 1 / sqrt(3) = h / (x + 40) => x + 40 = h * sqrt(3) ---- (equation 2)
Step 3: Now we have a system of two equations. We can solve it by substituting equation 1 into equation 2:
h / sqrt(3) + 40 = h * sqrt(3) => h * sqrt(
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