Answer:
The shortest possible length for the service road is 2.88 miles.
Step-by-step explanation:
According to the below diagram, [tex]A, B[/tex] and [tex]C[/tex] are the positions of airport, shopping center and factory respectively.
Given that, [tex]AB= 3.6 miles, BC= 4.8 miles[/tex] and [tex]AC= 6.0 miles[/tex]
In right triangle [tex]ABC[/tex]
[tex]tan(\angle ACB)=\frac{AB}{BC} \\ \\ tan(\angle ACB)= \frac{3.6}{4.8}=0.75\\ \\ \angle ACB= tan^-^1(0.75)=36.8698.... degree[/tex]
The shortest possible length for the service road from the shopping center to the highway that connects the airport and factory is [tex]BD[/tex].
That means, [tex]\triangle BCD[/tex] is also a right triangle in which [tex]\angle BDC=90\°[/tex], Hypotenuse[tex](BC)= 4.8[/tex] miles and [tex]BD[/tex] is the opposite side in respect of [tex]\angle DCB[/tex] or [tex]\angle ACB[/tex].
Now in right triangle [tex]BCD[/tex]
[tex]Sin(\angle ACB)=\frac{BD}{BC}\\ \\ Sin(36.8698...)=\frac{BD}{4.8}\\ \\ BD=4.8*Sin(36.8698...)=2.88[/tex]
So, the shortest possible length for the service road is 2.88 miles.
