Question

1. a surveyor must determine the distance, ab, across a river. he stands at point c, downriver 500 m from b, and using his theodolite, measures the angle of vision to a as 28°. how wide is the river?

Explanation:

Step1: Identify the triangle type

We have a right triangle \( \triangle ABC \) with \( \angle B = 90^\circ \), \( BC = 500 \) m (adjacent side to \( \angle C \)), and we need to find \( AB \) (opposite side to \( \angle C \)).

Step2: Use the tangent function

The tangent of an angle in a right triangle is defined as \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 28^\circ \), opposite side is \( AB \), and adjacent side is \( BC = 500 \) m. So, \( \tan(28^\circ)=\frac{AB}{500} \).

Step3: Solve for \( AB \)

Multiply both sides by 500: \( AB = 500\times\tan(28^\circ) \). Calculate \( \tan(28^\circ)\approx0.5317 \), then \( AB\approx500\times0.5317 = 265.85 \) m.

Answer:

The width of the river (distance \( AB \)) is approximately \( \boldsymbol{266} \) m (or more precisely \( 265.85 \) m).