Solve the problem.
-To find the distance $\mathrm { AB }$ across a river, a distance $\mathrm { BC } = 1101 \mathrm {~m}$ is laid off on one side of the river. It is found that $\mathrm { B } = 112.8 ^ { \circ }$ and $\mathrm { C } = 12.0 ^ { \circ }$. Find $\mathrm { AB }$ rounded to the nearest meter.
A) $230 \mathrm {~m}$
B) $279 \mathrm {~m}$
C) $282 \mathrm {~m}$
D) $233 \mathrm {~m}$

Question

Quizplus · Accepted Answer

To find the distance AB across the river, we can use the Law of Sines in the triangle ABC. 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. Thus, we have:$\frac{AB}{\sin C} = \frac{BC}{\sin A}$Given that $BC = 1101 \, \text{m}$, $B = 112.8^\circ$, and $C = 12.0^\circ$, we first need to find angle A. Since the sum of angles in a triangle is $180^\circ$, we have:$A = 180^\circ - B - C = 180^\circ - 112.8^\circ - 12.0^\circ = 55.2^\circ$Now, we can use the Law of Sines to find AB:$\frac{AB}{\sin(12.0^\circ)} = \frac{1101}{\sin(55.2^\circ)}$Solving for AB gives:$AB = \frac{1101 \times \sin(12.0^\circ)}{\sin(55.2^\circ)}$Using a calculator:$AB \approx 279 \, \text{m}$Therefore, the distance AB across the river, rounded to the nearest meter, is approximately 279 meters.

Solve the Problem AB\mathrm { AB }AB Across a River, a Distance

Solve the Problem $\mathrm { AB }$ Across a River, a Distance