Question

1. biology a biologist wants to know the width ( w ) of a river. from point ( a ), the biologist walks downstream 100 feet and sights to point ( c ) (see figure). from this sighting, it is determined that ( \theta = 54^circ ). how wide is the river?

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the adjacent side to the angle \(\theta = 54^\circ\) is 100 feet (the distance walked downstream), and the opposite side is the width \(w\) of the river. The tangent function relates the opposite and adjacent sides in a right triangle, so \(\tan\theta=\frac{w}{100}\).

Step2: Solve for \(w\)

We know \(\theta = 54^\circ\), so we can rewrite the formula as \(w = 100\times\tan(54^\circ)\). Using a calculator to find \(\tan(54^\circ)\approx1.3764\), then \(w = 100\times1.3764 = 137.64\) feet.

Answer:

The width of the river is approximately \(\boldsymbol{137.64}\) feet.