two students stand 1 yard apart and measure their respective angles of elevation to the top of a tree…

two students stand 1 yard apart and measure their respective angles of elevation to the top of a tree. student a measures the angle to be 57°, and student b measures the angle to be 46°. what is h, the height of the tree? use the law of sines to first find at. then use that measure to find the value of h. 3.0 yards 3.2 yards 3.8 yards 4.4 yards law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

two students stand 1 yard apart and measure their respective angles of elevation to the top of a tree. student a measures the angle to be 57°, and student b measures the angle to be 46°. what is h, the height of the tree? use the law of sines to first find at. then use that measure to find the value of h. 3.0 yards 3.2 yards 3.8 yards 4.4 yards law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Answer

Explanation:

Step1: Find angle $\angle ATB$

In $\triangle ATB$, the sum of interior - angles of a triangle is $180^{\circ}$. So, $\angle ATB=180^{\circ}-(57^{\circ}+46^{\circ}) = 77^{\circ}$.

Step2: Use the law of sines to find $AT$

In $\triangle ATB$, by the law of sines $\frac{AB}{\sin\angle ATB}=\frac{AT}{\sin B}$. Given $AB = 1$ yard, $\angle ATB = 77^{\circ}$, and $B = 46^{\circ}$. So, $AT=\frac{AB\times\sin B}{\sin\angle ATB}=\frac{1\times\sin46^{\circ}}{\sin77^{\circ}}$. Since $\sin46^{\circ}\approx0.7193$ and $\sin77^{\circ}\approx0.9744$, then $AT=\frac{0.7193}{0.9744}\approx0.7382$ yards.

Step3: Find the height $h$ of the tree

In right - triangle $AGT$, $\sin A=\frac{h}{AT}$. Since $A = 57^{\circ}$ and $AT\approx0.7382$ yards, then $h = AT\times\sin A$. $\sin57^{\circ}\approx0.8387$, so $h=0.7382\times0.8387\approx3.0$ yards.

Answer:

3.0 yards