Question

A and B are two balloons at the same height and 1000m apart. O is ground observation position,angle AOX=14 degrees 15 minutes and angle BOX=10 degrees 30 minutes. Find the height of the balloons.

Answer 1

To find the height of the balloons, we can first find the distance of each balloon from the observation point O using trigonometry.

Let the height of each balloon be h. Using trigonometry, we can write the following equations:

For balloon A:
tan 14°15' = h/x
x = h / tan 14°15' -----(1)

For balloon B:
tan 10°30' = h / (1000 - x)
1000 - x = h / tan 10°30'
x = 1000 - h / tan 10°30' -----(2)

Using equations (1) and (2), we can eliminate x by setting them equal to each other:

h / tan 14°15' = 1000 - h / tan 10°30'

Now, we can solve for h:

h * (1/tan 14°15' + 1/tan 10°30') = 1000
h = 1000 / (1/tan 14°15' + 1/tan 10°30')

Calculating the values of tan 14°15' and tan 10°30', we get:
tan 14°15' = tan(14 + 15/60)° = tan(869') = tan 869' ≈ 0.251
tan 10°30' = tan(10 + 30/60)° = tan(10.5°) ≈ tan 10.5° ≈ 0.188

Therefore, we have:
h = 1000 / (1/0.251 + 1/0.188)
h = 1000 / (3.984 + 5.319)
h = 1000 / 9.303
h ≈ 107.51 m

Therefore, the height of each balloon is approximately 107.51 meters.