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 = \square yd (round the answer to the nearest whole number.)

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle $\triangle ABC$, right-angled at $C$. We know $\angle A = 50^\circ$, adjacent side $AC = 510$ yds, and need opposite side $a = BC$. Use tangent:
$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step2: Substitute known values

Substitute $\theta = 50^\circ$, adjacent $= 510$:
$\tan(50^\circ) = \frac{a}{510}$

Step3: Solve for $a$

Rearrange to isolate $a$:
$a = 510 \times \tan(50^\circ)$
Calculate $\tan(50^\circ) \approx 1.191753592$, so:
$a \approx 510 \times 1.191753592$

Step4: Compute and round

Calculate the product and round to whole number:
$a \approx 607.794332$

Answer:

608 yd