Two girls are standing 100 feet apart. They both see a beautiful seagull in the air between them. The angles of elevation from the girls to the bird are 20° and 45°, respectively. How high up is the seagull? Round to three decimal places. Show your work.
To find the height of the seagull, we can use trigonometry and the angles of elevation provided. Let's call the height of the seagull "h".
We can start by drawing a diagram:
```
A
|\
h | \
| \
| \ 20°
| \
-------|-----\
100| \|
B----- C
45°
```
Here, A represents the location of the seagull, B and C represent the positions of the two girls, and h represents the height of the seagull.
We know that angle BAC is 45° and angle ABC is 20°. Additionally, we have the distance between the two girls, which is 100 feet.
Now, let's focus on triangle ABC. The tangent function can be useful here because we have opposite (h) and adjacent (100) sides. We can set up the following equation:
tan(20°) = h / 100
To solve for h, we can rearrange the equation:
h = tan(20°) * 100
Using a calculator, we can find the tangent of 20° to be approximately 0.36397:
h ≈ 0.36397 * 100
h ≈ 36.397
Therefore, the height of the seagull is approximately 36.397 feet.
draw a diagram, and you can see that
h cot20° + h cot45° = 100