Mathematics: 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.

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Civil an airport, a factory, and a shopping center

Ответ:

bihanna1234

10.03.2022

Mathematics

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

Step-by-step explanation:

According to the below diagram, A, B and 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 .

That means, $\triangle BCD$ is also a right triangle in which $\angle BDC=90\°$ , Hypotenuse (BC)= 4.8 miles and 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.

Civil an airport, a factory, and a shopping center are at the vertices of a right triangle formed by

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