Two men on the same side of a tall building notice the angle of elevation to the top of the building to be 30 dgrees and 60 dgrees respectively. If the height of the building is known to be h equals 60 m, find the distance (in meters)btween the 2 men.
60√3 m
To find the distance between the two men, we can use trigonometry and consider the right triangles formed by each man and the building.
Let's assume the distance between the two men is d. We can create a diagram to visualize the situation:
A
/ | \
h / | \
/ | \
/______|_______\
B d C d D
In the diagram above:
- A and B represent the two men.
- C represents the top of the building.
- AD and BD represent the distances from each man to the base of the building.
- CD represents the height of the building, given as h = 60 m.
Based on the given information, we know that angle BAC is 30 degrees and angle BCA is 90 degrees. Similarly, angle BAD is 60 degrees and angle BDA is 90 degrees.
Now, let's focus on triangle ABC. We can use the tangent function to find the length of AD:
tan(angle BAC) = opposite/adjacent
tan(30) = h / AD
1/sqrt(3) = 60 / AD
AD = 60 / (1/sqrt(3))
AD = 60 * sqrt(3)
Now, let's focus on triangle BCD. We can use the tangent function again to find the length of BD:
tan(angle BAD) = opposite/adjacent
tan(60) = h / BD
sqrt(3) = 60 / BD
BD = 60 / sqrt(3)
BD = 20 * sqrt(3)
Since the total distance between the two men is d, we have:
d = AD + BD
d = 60 * sqrt(3) + 20 * sqrt(3)
d = 80 * sqrt(3)
Therefore, the distance between the two men is approximately 138.56 meters.