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 \( \angle A = 30^\circ \), \( AC = 690 \) yds (adjacent side to \( \angle A \)), and \( BC = a \) (opposite side to \( \angle A \)). We use the tangent function, which is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). So, \( \tan(30^\circ)=\frac{a}{690} \).

Step2: Solve for \( a \)

We know that \( \tan(30^\circ)=\frac{1}{\sqrt{3}}\approx0.5774 \). So, we can rewrite the equation as \( a = 690\times\tan(30^\circ) \). Substituting the value of \( \tan(30^\circ) \), we get \( a = 690\times\frac{1}{\sqrt{3}} \). Rationalizing the denominator, \( a=\frac{690\sqrt{3}}{3}=230\sqrt{3}\approx230\times1.732 = 398.36 \). Rounding to the nearest whole number, \( a\approx398 \).

Answer:

\( 398 \)