To find the distance across a river a surveyor chooses points $A$ and $B$, $50$M apart and a reference point $C$On the opposite side of the river. Approximate the distance, $b$, between $A$And $C$Given that $\angle \mathrm { A } = 5 4^ { \circ }$And $\angle B = 77 ^ { \circ }$
A) $54.6 \mathrm {~m}$
B) $77.6 \mathrm {~m}$
C) $64.6 \mathrm {~m}$
D) $45.6 \mathrm {~m}$
E) $85.6 \mathrm {~m}$

Question

Quizplus · Accepted Answer

To solve this problem, we can use the law of sines in the triangle formed by points $A$, $B$, and $C$. The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Given:- $AB = 50$m- $\angle A = 54^\circ$- $\angle B = 77^\circ$First, we find $\angle C$ using the fact that the sum of angles in a triangle is $180^\circ$:$\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 54^\circ - 77^\circ = 49^\circ$Then, using the law of sines:$$\frac{AC}{\sin B} = \frac{AB}{\sin C}$$Solving for $AC$ (distance $b$):$$AC = \frac{AB \cdot \sin B}{\sin C} = \frac{50 \cdot \sin 77^\circ}{\sin 49^\circ}$$Calculating this gives a distance of approximately $64.6$m, making the correct answer $C$.

To Find the Distance Across a River a Surveyor Chooses AAA

To Find the Distance Across a River a Surveyor Chooses $A$