During a nature trip, Joni, Billy and Larry documented the location of an eagles nest. From point a, the campers observed the Eagles best on tip of a tree directly across a deep stream. The angle of elevation to the nest from point a is 35 degrees. The campers walked 42 m downstream to point b and observed that the straight line to the base of the tree made an angle of 30 degrees with the path along the bank of the river. How high is the eagles nest above the ground?
h / Da = tan(35º)
Da / 42 m = tan(30º)
h = 42 m * tan(35º) * tan(30º)
To find the height of the eagle's nest above the ground, we can break down the problem into two separate right-angled triangles and use trigonometric ratios.
Let's label the given information on a diagram:
C (Eagle's Nest)
|\
| \
| \
| \
35° | \ h (height)
| \
|______\
A x B (Tree)
From point A, the angle of elevation to the nest is 35 degrees. We need to find the height (h).
In right triangle ABC, we have:
Angle B = 90 degrees
Angle C = 180 - 90 - 35 = 55 degrees (sum of angles in a triangle is 180 degrees)
We can use the trigonometric ratio tangent (tan) to find h:
tan(35 degrees) = h / x
Now, let's look at the second observation at point B:
C (Eagle's Nest)
|\
| \
| \
30° | \
| \
| \ h (height)
| \
|_______\
A y B (Tree)
Here, the angle between the path along the river bank and the line to the base of the tree is 30 degrees. We also walked 42m downstream from point A to point B.
In right triangle ABC, angle B is still 90 degrees, but now we have a new angle, the complement of angle B (90 degrees - 30 degrees = 60 degrees).
We can again use the tangent (tan) ratio to find h:
tan(60 degrees) = h / y
Now, let's solve the two equations to find the value of h.
From the first equation:
tan(35 degrees) = h / x
h = x * tan(35 degrees)
From the second equation:
tan(60 degrees) = h / y
h = y * tan(60 degrees)
Since we are solving for the same height (h), we can equate the two expressions for h:
x * tan(35 degrees) = y * tan(60 degrees)
We know that x = y + 42m (as we walked 42m downstream from A to B).
Substituting this value:
(y + 42) * tan(35 degrees) = y * tan(60 degrees)
Now, solve this equation for y:
(y * tan(35 degrees)) + (42 * tan(35 degrees)) = y * tan(60 degrees)
Simplifying further,
(y * tan(35 degrees)) - (y * tan(60 degrees)) = - (42 * tan(35 degrees))
Factor out y:
y * (tan(35 degrees) - tan(60 degrees)) = - (42 * tan(35 degrees))
Now, divide both sides by (tan(35 degrees) - tan(60 degrees)):
y = - (42 * tan(35 degrees)) / (tan(35 degrees) - tan(60 degrees))
Once you calculate the value of y, you can substitute it back into any of the equations for h to find the height of the eagle's nest above the ground.