weather balloon is directly west of two observing stations that are 10 mi apart. The angles of elevation of the balloon from the two stations are 17.6 degrees and 78.2 degrees. How high is the balloon?

Answer 1

Make the following diagram
Draw a base line , label the balloon as B and above the base line
Label the point directly below the balloon A on the base line
label the two stations as C and D so that
angle ACB = 78.2 and angle ADB = 17.6
Now in triangle BCD , CD = 10 ,
angle BCD = 101.8 , (exterior angle theorem)
and angle CBD = 60.6° (angle sum of triangle theorem)

Using the sine law BC/sin17.6 = 10/sin60.6
you can find BC, which is then the hypotenuse of
triangle ABC
then use
sin 78.2 = height/BC to find the height