Question

The angle of depression of one side of a lake, measured from from a balloon 2500 feet above the lake is 43 degrees. The angle of depression to the opposite side of the lake is 27 degrees. Find the width of the lake.

Answer 1

Draw a picture of a balloon in the air and a lake in the distance.

Answer 2

Let the line joining the balloon's observer and his nadir point on the lake be x. The nadir point is that point on the lake that is directly below the balloon's observer. x is normal to the lake's surface, that is, makes a 90 degree angle with the lake's surface.

Let y be the join between the observer and the first side of the lake. y is the hypotenuse of a right triangle. The distance between the nadir and the lakes edge is:

x Cot[43 Degrees ] = 2500 Cot[Pi 43/180]

In a similar manner the distance between the nadir and the opposite side of the lake is:

2500 Cot[Pi 27/180]

The sum of the two cotangent expressions is the distance across the lake to the nearest foot:

2500 Cot[Pi 43/180] + 2500 Cot[Pi 27/180]=
\[2500 \left(\text{Cot}\left[\frac{43 \pi }{180}\right]+\text{Cot}\left[\frac{3 \pi }{20}\right]\right)=7587 \]

Answer 3

Thank You! Please help me figure this out. The angle of elevation to the top of the radio atenna on the top of a building is 53.4 degrees. After moving 200 feet closer to the building, the angle of elevation is 63.4 degrees. Find the height of the building if the height of the atenna is 189 feet. Please HELP!

Answer 4

Give me a few minutes to look at it.

Answer 5

Got the equations, want to check out the answers.

Answer 6

This is the first time I have tried to solve a problem in a face book type of environment with an audience.

Let x be the distance from the 63 degree measurement position to the base of the building. Now back up 200 feet and you are x+200 feet from the building's base.
Let y be the height of the building. Then the top of the antenna is y +189 above the ground.

What you have is two equations in two unknowns.

The tangent of 53 degrees is (y+189)/(x+200).
The tangent of 63 degrees is (y+189)/x

The solution for y, the height of the building is:
\[y\to -189+\frac{200 \text{Cot}\left[\frac{3 \pi }{20}\right] \text{Cot}\left[\frac{37 \pi }{180}\right]}{\text{Cot}\left[\frac{3 \pi }{20}\right]-\text{Cot}\left[\frac{37 \pi}{180}\right]}=630.5 \]

Answer 7

Used Mathematica 8 to get you an exact symbolic solution for y.

Answer 8

Used Mathematica 8 to get you an exact symbolic solution for y.