While visiting the Sand Duns National Park in Colorado, Cole approximated the angle of elevation to the top of a sand dune to be 20 degrees. After walking 800 ft closer he guessed that the angle of elevation had increased by 15 degrees. Approximately how tall is the dune he was observing?....There is a picture in the book on page 452....it is a triangle with an obtuse angle...so should I draw a line to make it a right angle? I am not sure how to go about answering this??....thanks.
See also my answer on the angles of elevation and balloon problem, posted at the access page for Mth 164. This problem is identical in structure. The two explanations are worded a little differently, so one might make more sense than the other.
You can make two right triangles, which is a good thing because you have two unknowns (height and original distance from the dune), and you're going to need two equations.
This is a great question, and you're not the only one who asked it, so we'll be looking at it in class.
The first triangle has an unknown side and an angle of elevation of 20 degrees--that is, the angle between a line along the ground and a line to the top of the dune is 20 deg. The unknown side is the distance along the ground a point directly under the peak of the dune, so this side will be perpendicular to the height of the dune. So the unknown side will be the side adjacent to the 20 degree angle.
If we call the height of the dune h, and the unknown distance x, we therefore have
tan(20 deg) = h / x.
Remember that tan(20 deg) is just a number you can find on your calculator.
The second triangle has angle of elevation 35 deg. A triangle constructed in the same manner as the first will have altitude h and adjacent side x - 800 ft. So
tan(35 deg) = h / (x - 800 ft).
This gives us two equations we can solve for x and h. The easiest way to solve is to solve the first equation for x, then substitute this result in to the second equation. This will give you an equation you can solve for h.