Question

1. -/2 points a biologist wants to know the width ( w ) of a river to properly set instruments for an experiment. from point ( a ), the biologist walks downstream 100 feet and sights to point ( c ) (across the river). it is determined that ( \theta = 47^circ ). how wide is the river? (round your answer to one decimal place.)

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the adjacent side to the angle \( \theta = 47^\circ \) 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: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \).

Step2: Substitute the known values

Substitute \( \theta = 47^\circ \) and adjacent = 100 into the tangent formula: \( \tan(47^\circ)=\frac{w}{100} \).

Step3: Solve for \( w \)

Multiply both sides by 100 to solve for \( w \): \( w = 100\times\tan(47^\circ) \). Calculate \( \tan(47^\circ)\approx1.0724 \), so \( w = 100\times1.0724 = 107.24 \). Rounding to one decimal place gives \( w\approx107.2 \).

Answer:

107.2