Question

the people of bridgetown wanted to build a bridge across a nearby river. since they were poor swimmers, their master trigonomos agreed to measure the width of the river without actually crossing it.
trigonomos spotted a tree across the river and marked the spot directly across from it. then he walked to another point 15 meters down the river and found that the angle between his side of the river and the line connecting him to the tree was 78°.
what is the width of the river?
round your final answer to the nearest hundredth.
meters

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the adjacent side to the \(78^\circ\) angle is 15 meters (the distance walked along the river), and the opposite side is the width of the river (\(w\)) we need to find. We use the tangent function, which is \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up the equation

\(\tan(78^\circ)=\frac{w}{15}\)

Step3: Solve for \(w\)

Multiply both sides by 15: \(w = 15\times\tan(78^\circ)\)
Calculate \(\tan(78^\circ)\approx4.7046\), so \(w = 15\times4.7046 = 70.569\)

Step4: Round to the nearest hundredth

\(w\approx70.57\)

Answer:

70.57