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 =? 43° 540 yds a = □ yd (round the answer to the nearest whole number.)

Explanation:

Step1: Identify the trigonometric relationship

In right - triangle ABC, we know the adjacent side (AC = 540 yds) to the angle A = 43° and we want to find the opposite side (a). We use the tangent function since $\tan(A)=\frac{\text{opposite}}{\text{adjacent}}$.
$\tan(43^{\circ})=\frac{a}{540}$

Step2: Solve for a

Multiply both sides of the equation by 540: $a = 540\times\tan(43^{\circ})$.
We know that $\tan(43^{\circ})\approx0.932515$.
So, $a = 540\times0.932515$.
$a\approx503.5581$.

Step3: Round the answer

Rounding 503.5581 to the nearest whole number gives $a\approx504$.

Answer:

504