Question

to rebuild a bridge that was washed out by a flood, a surveyor needs to determine the width of the river. she has measured the distance from the start of the bridge to where she is standing to be 200 m. she sets up her transit and measures the angle to the point across the river where the bridge will end as 24°, as represented by the diagram below. 4. determine the width of the river to the nearest metre. hint: θ = ____ opposite = ________ adjacent = ________ trig ratio used: __________

Explanation:

Step1: Identify trigonometric ratio

We have a right - triangle. The angle \(\theta = 24^{\circ}\), the adjacent side \(adjacent = 200\) m (distance from the start of the bridge to the surveyor), and the width of the river is the opposite side (\(opposite\)) with respect to the angle \(\theta\). The trigonometric ratio that relates the opposite and adjacent sides is the tangent ratio. The formula for the tangent of an angle in a right - triangle is \(\tan\theta=\frac{opposite}{adjacent}\).

Step2: Solve for the opposite side

We know that \(\tan\theta=\frac{opposite}{adjacent}\), and we want to find \(opposite\). Rearranging the formula gives \(opposite = adjacent\times\tan\theta\). Substitute \(\theta = 24^{\circ}\) and \(adjacent = 200\) m. Using a calculator, \(\tan(24^{\circ})\approx0.4452\). Then \(opposite=200\times\tan(24^{\circ})\).

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Answer:

\(\theta = 24^{\circ}\), \(opposite=\text{width of the river}\), \(adjacent = 200\) m, \(trig\ ratio\ used=\tan\) and the width of the river is \(89\) m.