Question :The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it was 60°. Find the height of the tower.
The correct answer :Let’s assume that the height of the tower is represented by “h” meters and the initial length of the shadow when the sun’s altitude was 60 degrees is represented by “x” meters.
When the sun’s altitude changes to 30 degrees, the shadow length increases by 40 meters. Let’s call this new length “x + 40” meters.
We can use the tangent function to find the height of the tower:
tan(60) = h/x (since the angle of elevation is 60 degrees)
tan(30) = h/(x+40) (since the angle of elevation is 30 degrees and the length of the shadow has increased by 40 meters)
We can rewrite the first equation as:
x = h/tan(60)
And the second equation as:
x + 40 = h/tan(30)
Now we can substitute the value of “x” from the first equation into the second equation and solve for “h”:
h/tan(60) + 40 = h/tan(30)
h/√3 + 40 = h/1/√3
h/√3 + 40√3/3 = h
h(1/√3 – 1) = -40√3
h = 40√3/(1 – 1/√3)
h ≈ 69.28 meters (rounded to two decimal places)
Therefore, the height of the tower is approximately 69.28 meters.
