Question

Jason and Allison are hiking in the woods when they spot a rare owl in a tree. Jason stops and measures an angle of elevation of 22° 8’ 6”. At the same time, Allison is standing 48 feet closer to the tree, measuring an angle of elevation of 30° 40’ 30” to the owl. If Jason and Allison are the same height, their eyes 5 feet from the ground, find the height of owl in the tree.

Answer 1

Answer:

The owl is at a height of about 67.13 feet from the ground

Step-by-step explanation:

To start with, let's convert the given angles of elevation to decimal degrees:

$22^o\, 8' \,6" = 22 +\frac{8}{60} +\frac{6}{3600}=22.135^o\\30^o\, 40' \,30" = 30 +\frac{40}{60} +\frac{30}{3600}=30.675^o$

Now let's make a diagram of the two right angle triangles associated with the position of the hikers, the owl on the tree, and the angles of elevation at which each hiker sees the owl. Please see the attached image.

notice two right angle triangles (a green one and an orange one that have in common the height of the owl relative to the level of the hikers' eyes Labeled with the letter "T". There is also an unknown labeled "x" in the image, and that is the distance between the hiker closer to the tree, and the tree.

From those right angle triangles we can derive simple trigonometric equations involving the tangent of the known angles, the sides opposite to the angles (T in both triangles), and the side adjacent to the angle (x for the smaller triangle, and "x+48 ft" for the larger triangle).

These equations are:

$tan(22.135^o)=\frac{T}{48+x} \\tan(30.675^o)=\frac{T}{x}$

We can solve this system of two equations and two unknowns by solving for T in both and then equaling both expressions:

$T=(48+x)\,tan(22.135^o)\\T=x\,\,tan(30.675^o)\\\\(48+x)\,tan(22.135^o)=x\,\,tan(30.675^o)\\48\,tan(22.135^o)+x\,\,tan(22.135^o)=x\,\,tan(30.675^o)\\48\,tan(22.135^o)=x\,\,(tan(30.675^o)-tan(22.135^o))\\x=\frac{48\,tan(22.135^o)}{tan(30.675^o)-tan(22.135^o)} \\x=104.75\,\,ft$

Now we can find T:

$T-x\,\,tan(30.675^o)\\T=(104.75\,ft)\,\,tan(30.675^o)\\T=62.1337\,\,ft$

Notice now that this is the height the owl is relative to the eye level of the hikers, so if we want the height of the owl from the ground, we need to add 5 ft to this number:

Height of the owl from the ground: 62.1337 ft + 5 ft = 67.1337 ft