Answer:

Given:

Angle of elevation from point A = 18°

Angle of elevation from point B = 12°

Distance between points A and B = 8.5 miles

To find:

Height of the balloon in the following situation:

b) A and B are on the same sides of the balloon.

Solution:

Let h be the height of the balloon and x be the distance between the closer point of observation and the balloon (as shown in the diagram below).

We can set up the following system of equations:

tan(18°) = h/x

tan(12°) = h/(8.5 - x)

Solving for x in the second equation:

x = (8.5h)/(h/tan(12°) + tan(18°))

Substituting this into the first equation:

tan(18°) = h/[(8.5h)/(h/tan(12°) + tan(18°))]

tan(18°) = (h tan(12°) + h tan(18°))/8.5

8.5 tan(18°) = h tan(12°) + h tan(18°)

h = (8.5 tan(18°))/(tan(12°) + tan(18°))

Using a calculator, we get:

h ≈ 1.28 miles

Therefore, the height of the balloon when points A and B are on the same sides is approximately 1.28 miles.