Question

As seen in the accompanying diagram, a person cantravel from New York City to Buffalo by goingnorth 170 miles to Albany and then west 280 milesto Buffalo.Buffalo280 milesAlbany170 milesNew York CityIf an engineer wants to design a highway to connectNew York City directly to Buffalo, at what angle, x,would she need to build the highway? Find theangle to the nearest degree. To the nearest mile,how many miles would be saved by travelingdirectly from New York City to Buffalo rather thanby traveling first to Albany and then to Buffalo?

Answer 1

angle x=59º

122 miles would be saved

1) That engineer needs the angle to make that road, our hypotenuse. Then let's find the angle that can be found by using a trigonometric ratio, in this case, the tangent since there are two legs the opposite and the adjacent one, and then we'll make use of the arctangent to get the angle:

`\begin{gathered} \tan (x)=\text{ }(280)/(170) \\ x=\tan ^(-1)((280)/(170)) \\ x=58.73 \end{gathered}`

2) Since the shortest way from New York to Buffalo is through a straight line, let's find out the missing leg, in this case, the hypotenuse.

a² = b² + c²

a² = 170² +280² The hypotenuse is unknown

a² = 107300 Take the square root on both sides

√a²=√107300

a =327.56 approximately 328

3) Now let's subtract from the sum of the distance between those two legs:

New York - Albany: 170

Albany - Buffalo: 280

280 +170 = 450 miles

So, subtracting from 450, we have 450-328 = 122 miles would be saved.

4) Hence, the answers are:

She needs to build a road at an angle (x) of approximately 59º

122 miles would be saved from NYC -Buffalo