Hello, nathan02079!

Streelights A, B, C, D, and E are placed 50 m apart on the main road.

The light from a streetlight is effective up to a distance of 60 m.

(a) Find the distance from A to the farthest point on the side road that is illuminated.

(b) Determine the length of the side road that is illuminated by both lights C and D. Code:

F *
o
* \
* \
* \
* 21° \
o--------o--------o--------o-------o
A 50 B 50 C 50 D 50 E

From $\displaystyle E$, draw $\displaystyle EF$ perpendicular to the side road.

In right triangle $\displaystyle EFA\!:\;\;\sin21^o \:=\:\frac{EF}{200} \quad\Rightarrow\quad EF \:=\:200\sin21^o \:\approx\;72$ m.

Hence, streetlight $\displaystyle E$ does *not* effectively illuminate the side road.

Code:

G *
o
* /
* /
* / 60
* 21° /
o-------o-------o--------o
A 50 B 50 C 50 D

Draw $\displaystyle DG = 60$ so that $\displaystyle \angle D$ is obtuse.

Law of Sines: .$\displaystyle \frac{\sin G}{150} \:=\:\frac{\sin21^o}{60} \quad\Rightarrow\quad \sin G \:=\:0.895919874 \quad\Rightarrow\quad G \:\approx\:63.6^o$

. . Then: .$\displaystyle \angle D \:=\:180^o - 21^o - 63.6^o \:=\:95.4^o$

Law of Sines: .$\displaystyle \frac{AG}{\sin95.4^o} \:=\:\frac{60}{\sin21^o} \quad\Longrightarrow\quad AG \;\approx\;167.4\text{ m}$ .(a)