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
32°
600 yds
c
b
a = ?
a = \square yd
(round the answer to the nearest whole number.)

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\) (right - angled at \(C\)), we know the adjacent side to the angle \(32^{\circ}\) (which is \(AC = 600\) yd) and we need to find the opposite side \(a=BC\). The trigonometric ratio that relates the opposite side and the adjacent side of an angle in a right triangle is the tangent function. The formula for the tangent of an angle \(\theta\) in a right triangle is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Substitute the values into the tangent formula

Here, \(\theta = 32^{\circ}\), the adjacent side \(AC = 600\) yd, and the opposite side is \(a\). So we have \(\tan(32^{\circ})=\frac{a}{600}\).

Step3: Solve for \(a\)

To solve for \(a\), we can multiply both sides of the equation by \(600\). So \(a = 600\times\tan(32^{\circ})\).
We know that \(\tan(32^{\circ})\approx0.6249\) (using a calculator). Then \(a=600\times0.6249 = 374.94\).

Step4: Round to the nearest whole number

Rounding \(374.94\) to the nearest whole number gives \(375\).

Answer:

\(375\)