You are here:

Advanced Math/Angle of elevation

Advertisement

Expert: - 2/16/2009


Question
Two observers are 500 feet apart. Each measures the angle of elevation of a hot air balloon that lies in a vertical plane passing through their locations; the angles are 53 and 42. Find the height of the balloon if the observers are on the same side of the balloon. (Simplify the answer as much as possible.)

Answer
The angles are 53° and 42°.  Since the larger angle is 53, the angle
with the ballon and the other observer is 180-53=127.  Now we have
two angles of the triangle, 127° and 52°.

Since the sum of the angles of a triangle are 180°,
so 180° - 127° is 53°.  So the angle between the observers from
the view of the person in the airplane is 53°, but that's not that
important, it is only an exercise on finding the other angle in a
triangle.

What is really important is to form two triangles.
Both of them has height h.
One of them has base x and the others has base x+500.

We know that tan(53°) = h/x and the tan(42°) = h/(x+500).
Since the tan(53°) and than tan(42°) can be looked up
(or found on a calculator), the two unknowns are x and h.

It can be seen that h = x tan(53°), so putting this into the second
equation gives us tan(42°) = (x+tan(53°))/(x+500).

Here is how to solve the equation:

Multiplied both sides of the equation by (x+500).

Multiply the left side out by multiplying the x and the 500
each by tan(42°).

The right side has an x and the left side has a 500*tan(42°).
Subtract 500*tan(42°)+x from both sides.  It should cancel on
both sides and make both terms into negatives on the other side.

The equation now has two terms involving an x on the left side.
Factor out the x and divide both sides by the multiplying factor,
and there is the solution for x.

If you want to, the tan(53°) and tan(42°) can be evaluated on
a calculator.  Make sure the calculator is in degrees.  If it
is not, take the sin(90).  The answer should be 1.

Advanced Math

All Answers

Answers by Expert:

Ask Experts

Volunteer

Scott A Wilson

Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

Experience

Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people nearby, friends with math questions, and several people every day on the PC, and you're probably make one more.

©2012 About.com, a part of The New York Times Company. All rights reserved.