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 600 yds c 31° b a = ? (round the answer to the nearest whole number.) a = \square yd

Explanation:

Step1: Identify the trigonometric ratio

We have a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), \( AC = 600 \) yds, \( \angle BAC = 31^\circ \), and we need to find \( a = BC \). We use the tangent function, which is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \tan(31^\circ)=\frac{BC}{AC} \), so \( \tan(31^\circ)=\frac{a}{600} \).

Step2: Solve for \( a \)

Rearrange the formula to solve for \( a \): \( a = 600\times\tan(31^\circ) \). Using a calculator, \( \tan(31^\circ)\approx0.6009 \). Then \( a = 600\times0.6009 = 360.54 \), which rounds to 361.

Answer:

361