Let's go through the steps to solve the problem:
### Step 1: Draw a Picture
Draw a right triangle to represent the situation. The height of the waterfall is the vertical leg, the distance from the bear to the base of the waterfall is the horizontal leg, and the line of sight from the top of the waterfall to the bear forms the hypotenuse with a given angle of depression of 27°.
```
|
|\
| \
| \
| \
| \
|___27°\
height distance
```
### Step 2: Set up an Equation Using Trigonometry
We're dealing with a right triangle where we know:
- The height of the waterfall is the opposite side ('height') = 365 feet.
- The angle of depression is 27°.
- We need to find the adjacent side ('distance'), which is the distance from the base of the waterfall to the bear.
We can use the tangent function which relates the opposite side and the adjacent side in a right triangle:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here,
[tex]\[ \tan(27°) = \frac{365}{\text{distance}} \][/tex]
### Step 3: Solve for the Distance
Rearrange the equation to solve for distance:
[tex]\[ \text{distance} = \frac{365}{\tan(27°)} \][/tex]
When you calculate this expression, you would get:
[tex]\[ \text{distance} \approx \frac{365}{0.5095254494944288} \approx 716 \text{ feet} \][/tex]
### Final Answer
To the nearest foot, the bear is approximately 716 feet from the base of the waterfall.
