Answer:
the length AB across the lake is approximately 95.08 meters.
Step-by-step explanation:
To determine the length AB across the lake, we can use the law of sines. According to the law of sines, the ratio of the length of a side of a triangle to the sine of its opposite angle is constant.
In triangle ABC:
angle C = 32°
angle B = 58°
length AC = 50 m
Let's find angle A:
angle A = 180° - angle B - angle C
angle A = 180° - 58° - 32°
angle A = 90°
Now, we can use the law of sines:
AB / sin(A) = AC / sin(C)
Substituting the values:
AB / sin(90°) = 50 m / sin(32°)
Since sin(90°) = 1, the equation simplifies to:
AB = 50 m / sin(32°)
Let's calculate the length AB:
AB = 50 m / sin(32°)
AB ≈ 95.08 m
Therefore, the length AB across the lake is approximately 95.08 meters.