Question

A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are 20° and 22°. How high is the balloon?

Answer 1

Answer:3.677 miles

Step-by-step explanation:

First we have to know that the distance between two consecutive milestone is 1mile, I have attached a diagram representing the problem, point C being the point of the balloon and point A and point B being the points of the consecutive milestones

I labelled the horizontal length between the balloon and the first milestone x and the height of the balloon h

From the diagram, we use the tan rule for right angle triangles to find x and h

Tan22=h/x

Making her the subject of formula

h=xtan22 (1)

Similarly

Tan20=h/(x+1)

h=(x+1)tan20 (2)

Putting (1) and (2) together

We solve for x, and then input our value for in (1) to get the value for h, the solution is in the second picture I sent

Note: tan rule says the tan of an angle is equal to the opposite side of the triangle(side directly facing the angle) divided by the adjacent side of the triangle( side of the triangle the angle is resting on)

Angle of depression from C to B is also angle of elevation from B to C( alternate angle are equal)