Question

Civil an airport, a factory, and a shopping center are at the vertices of a right triangle formed by three highways. the airport and factory are 6.0 miles apart. their distances from the shopping center are 3.6 miles and 4.8 miles, respectively. a service road will be constructed from the shopping center to the highway that connects the airport and factory. what is the shortest possible length for the service road? round to the nearest hundredth.

Answer 1

The shortest possible length for the service road is 2.88 miles.

Explanation:

According to the below diagram,
`A, B` and
`C` are the positions of airport, shopping center and factory respectively.

Given that,
`AB= 3.6 miles, BC= 4.8 miles` and
`AC= 6.0 miles`

In right triangle
`ABC`

`tan(\angle ACB)=(AB)/(BC) \\ \\ tan(\angle ACB)= (3.6)/(4.8)=0.75\\ \\ \angle ACB= tan^-^1(0.75)=36.8698.... degree`

The shortest possible length for the service road from the shopping center to the highway that connects the airport and factory is
`BD`.

That means,
`\triangle BCD` is also a right triangle in which
`\angle BDC=90\°`, Hypotenuse
`(BC)= 4.8` miles and
`BD` is the opposite side in respect of
`\angle DCB` or
`\angle ACB`.

Now in right triangle
`BCD`

`Sin(\angle ACB)=(BD)/(BC)\\ \\ Sin(36.8698...)=(BD)/(4.8)\\ \\ BD=4.8*Sin(36.8698...)=2.88`

So, the shortest possible length for the service road is 2.88 miles.