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Romashka-Z-Leto [24]
3 years ago
9

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.

Mathematics
1 answer:
vekshin13 years ago
4 0

Answer:

<em>The shortest possible length for the service road is 2.88 miles.</em>

Step-by-step 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)=\frac{AB}{BC} \\ \\ tan(\angle ACB)= \frac{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)=\frac{BD}{BC}\\ \\ Sin(36.8698...)=\frac{BD}{4.8}\\ \\ BD=4.8*Sin(36.8698...)=2.88

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

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