Two men on the same side of a tall building notice the angle of elevation to the top of the building to be 30o and 60o respectively. If the height of the building is known to be h =120 m, find the distance (in meters) between the two men.
Assuming that both observers are at an elevation level with the base of the building.
Height of the building, H = 120m
Distance of observer 1 = D1
Angle of elevation = α
By the definition of tangent = opp/adj
H/D1=tan(α), therefore
D1=H/tan(α)
Since H and α are known, D1 can be calculated numerically.
Distance of observer 2 = D2
Angle of elevation = β
H/D2=tan(β)
D2=H/tan(β)
Distance between observers = D2-D1
To find the distance between the two men, we can use trigonometric ratios. Let's label the distance between the two men as "x". We can form a right triangle with the height of the building as the vertical side, the distance between the first man and the building as the base, and the distance between the second man and the building as the hypotenuse.
First, let's find the height of the triangle formed by the first man and the building. We have the angle of elevation as 30 degrees and the height of the building as 120 meters. Using the trigonometric ratio for tangent (tan), we have:
tan(30°) = Opposite/Adjacent
tan(30°) = h/x
Rearranging the equation, we have:
x = h/tan(30°)
x = 120/tan(30°)
Now, let's find the height of the triangle formed by the second man and the building. We have the angle of elevation as 60 degrees and the height of the building as 120 meters. Using the same trigonometric ratio for tangent (tan), we have:
tan(60°) = Opposite/Adjacent
tan(60°) = h/(x + 120)
Rearranging the equation, we have:
x + 120 = h/tan(60°)
x + 120 = 120/tan(60°)
Now, we can solve for x. Let's plug in the values for h and calculate:
x + 120 = 120/tan(60°)
x + 120 = 120/√3
x + 120 = 40√3
Subtracting 120 from both sides:
x = 40√3 - 120
Calculating the value:
x ≈ -11.09 m
Since distance cannot be negative, we can ignore the negative value. Therefore, the distance between the two men is approximately 11.09 meters.