Trig Word Problem

[Guest]

A hot air balloon in between two spotters who are 1.2 miles apart. One spotter reports that the angle of elevation of the balloon is 76 degrees, and the other reports that the angle of elevation of the balloon is 68 degrees. What is the altitude of the balloon in miles?


So I can easily find all of the angles of the triangle and those of the right triangle made by the line drawn from the balloon to the ground, which will eventually be the answer to the altitude. I just don't know how to find any other lengths to figure this out. Thanks in advance.

 

[stapel]

Draw the baseline between the two spotters, and label as "1.2".

Draw the height line, between the two spotters, to the balloon, and label as "h".

Draw the lines of sight from the spotters, and label with the appropriate angle measures.

Label the distance between one of the spotters and the height line as "x"; label the other distance as "1.2 - x".

Use the triangles to create a system of two equations in two unknowns. Solve for "h".

If you get stuck, please reply showing how far you have gotten in following the instructions. Thank you.

Eliz.

 

[TchrQbic]

A hot air balloon in between two spotters who are 1.2 miles apart. One spotter reports that the angle of elevation of the balloon is 76 degrees, and the other reports that the angle of elevation of the balloon is 68 degrees. What is the altitude of the balloon in miles?


So I can easily find all of the angles of the triangle and those of the right triangle made by the line drawn from the balloon to the ground, which will eventually be the answer to the altitude. I just don't know how to find any other lengths to figure this out. Thanks in advance.

From the angles of elevation given, one spotter is y miles from the right angle directly below the balloon. The other spotter is y + 1.2 miles away from directly below. Let x be the altitude of the balloon. Then you have for the closer spotter tan 76 = x/y. For the other spotter, tan 68 = x/(y+1.2)

Solve each of those for x and then equate them since x represents the altitude in both. You must find y first, then you can find x.

 

[galactus]

Another way is to use the law of sines:

\(\displaystyle \frac{a}{sin(A)}=\frac{b}{sin(B)}\)

\(\displaystyle \frac{1.2}{sin(A)}sin(B)=b\)

This will give the length of side b.

Or we can use:

\(\displaystyle \frac{a}{sin(A)}=\frac{c}{sin(C)}\)

\(\displaystyle \frac{1.2sin(C)}{sin(A)}=c\)

Now, use law of sines again.

\(\displaystyle \frac{1.2sin(B)}{sin(A)}sin(C)\)

Since we have three angles and a side, we have a formula:

\(\displaystyle \L\\\frac{1.2sin(B)sin(C)}{sin(A)}=h\)