Question

to find the distance across a lake, a surveyor took the measurements in the figure shown. use these measurements to determine how far it is across the lake. a = □ yd (round the answer to the nearest whole number.)

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle \( \triangle ABC \) with \( \angle A = 32^\circ \), \( AC = 690 \) yd (adjacent side to \( \angle A \)), and \( BC = a \) (opposite side to \( \angle A \)). We use the tangent function, which is \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Step2: Substitute the values

So, \( \tan(32^\circ) = \frac{a}{690} \). To solve for \( a \), we multiply both sides by 690: \( a = 690 \times \tan(32^\circ) \).

Step3: Calculate the value

Using a calculator to find \( \tan(32^\circ) \approx 0.6249 \). Then, \( a \approx 690 \times 0.6249 \approx 431.181 \). Rounding to the nearest whole number gives \( a \approx 431 \).

Answer:

431