Question

Two people located 500 yards apart have spotted a hot air balloon. The angle of elevation from one person to the

balloon is 67°. From the second person to the balloon the angle of elevation is 46°. How high is the balloon when it
is spotted?

Answer 1

`h =√(390.731^2 -152.671^2)=359.670 ft`

Explanation:

The situation is illustrated on the figure attached.

We can begin finding the values for h1 and h2 and in order to do this we can use the sine law.

`(sin (67))/(h_2) = (sin (67))/(500)`

From this we have that
`h_2 = 500`

And for h1 we have this:

`(sin(67))/(500) = (sin(46))/(h_1)`

And we got
`h_1 = (sin(46))/(sin(67)) 500= 390.731 ft`

Now we cna use the Pythagorean identity, since we have two right triangles. If we apply this identity to the right triangle on the left we have this:

`h^2 + x^2 = h^2_1` (1)

And for the right triangle we got:

`h^2 +(500-x)^2 = h^2_2` (2)

We can subctract equation (2) and (1) and we got:

`(500-x)^2 -x^2 = h^2_2 -h^2_1`

And if we apply some algebra we got this:

`250000 -1000 x +x^2 -x^2 = 97329.185`

`250000 -1000 x = 97329.185`

`1000 x = 152670.815`

`x =152.671`

Now since we have the value of x we can find the value for h like this:

`h^2 = h^2_1 -x^2`

`h =√(390.731^2 -152.671^2)=359.670 ft`