A biologist wants to know the width w of a river so that instruments for studying the pollutants in the water can be set properly. From point A, the biologist walks downstream 100 feet and sights to point C. From the sighting, it is determined that theta = 54°. How wide is the river? Round to the nearest tenth.
Poor wording. Where is C?
I will assume that C is on the other side of the river directly across from A
and theta is the angle the river forms with the line to C from his destination.
so AC/100 to tan 54°
AC = 100tan54 = ....
To find the width of the river, we can use trigonometry and the information provided in the problem.
Let's draw a diagram to visualize the situation:
```
A
/ |
/ | w
/ |
/ θ|
/____|
C 100 ft
```
Here, we have a right triangle formed by points A, C, and a point directly below A (let's call it D). The width of the river is represented by the segment CD.
Using trigonometry, we can use the tangent function in this case because we have the opposite (CD) and adjacent (100 ft) sides of the angle θ.
The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
So, we have:
tan(θ) = opposite/adjacent
tan(54°) = CD/100 ft
Now, we can solve for CD (the width of the river):
CD = tan(54°) * 100 ft
Calculating this using a calculator or a programming tool, we find:
CD ≈ 115.37 ft
Therefore, the width of the river (CD) is approximately 115.4 feet (rounded to the nearest tenth).