trigonometry question?

Question

from a balloon 2500 ft. high, a command post is seen with an angle of depression of 77 degrees. How far is it from a point on the ground below the balloon to the command post, to the nearest tenth of a mile? the book says the answers is 3.5 miles but i dont know how to get that answer.

Answer 1

You basically have a right triangle with the baloon at the top, like this:
* /|
* H / |
* / | h
* /_ |
* x

The height, 2500 ft, is H; the distance you're looking for is x. The angle of depression is the angle that the observers look down from the horizontal, so the angle at the top of the triangle as drawn is (90-77) degrees, or 13 degrees.You know that cos (13 deg) = h/H, or H = h/cos(13 deg).
So sin(13 deg) = x/H, or x = H sin(13 deg) = [h/cos(13 deg)] sin(13 deg). Plug in the numbers and you get the answer. Don't forget that h is given in feet, so you have to convert yhe answer to miles by dividing by 5280 feet per mile.

Answer 2

Look at this diagram, which oddly enough uses a hot air balloon:
http://www.mathwords.com/a/angle_depression.htm

Draw a rectangle. Connect two of the corners. The top corner with the diagonal is where the balloon is. The bottom corner with the diagonal is where the fort is. Label these. The diagonal makes an angle of 77 degrees with the top line. Label this too.

The height of the rectangle is the height of the balloon, or 2500 feet. We want to find out the bottom length of the rectangle, so label it "x". Notce though that the diagonal splits the rectangle into two right-triangles, the bottom triangle of which as a side length of 2500, 13 degrees for the top angle, and "x" for the length of the bottom. You'll notice that tan(13) = x/2500. So x = 2500*tan(13). This is in feet, so you need to divide by 5280 to get miles.

I don't get 3.5 miles though. I tried it with 13 being the original angle of depression, and that doesn't work either. Are you sure the book says 3.5 miles?

Answer 3

I get 2.051mi if you got that too, I would say that the book is wrong. What I did was drew a right triangle. The top, point A, is where the balloon is. Then, point B directly below. The distanct AB you know is 2500ft. And finally, point C the command post.

I used TAN of angle BAC equals BC divided by AB.
so TAN (77) = x/2500 ===> x = 2500*tan(77)
Divide by 5280 for miles, but you'll get 2.05 not 3.5. So, I'm wrong or the book is.

Answer 4

On an x-y coordinate system plot the Point B(0, 2500). Label the origin O(0,0). Draw a line through B parallel to the x-axis to a point C. Now draw a line from C to the x-axis at F such that the angle CBF = 77 degrees. This is the angle of depression. CF is the distance from directly below the baloon to the command post.

Since BC || CF angle BFC = 77 degrees.

Thus 2500/ CF = tan 77
CF = 2500/tan 77
CF = 577.2 feet.

Your book is screwed up.

Answer 5

You have a right triangle. The opposite leg is 2500 ft. The angle is 77º. You want to find the adjacent leg.

tan Θ = opp/adj
tan 77º = 2500/adj
adj = 2500/tan 77º

I don't get 3.5 miles either, by the way.