Question

Starting from an airport, an airplane flies 210 miles southeast and then 210 miles south. How far, in miles, from the airport is the plane? (Round your answer to the nearest mile.)

Answer 1

The plane is 388 miles far from the airport.

Explanation:

We know that, the angle between southeast and south directions is
`135^\circ`.

The plane travels as per the triangle as shown in the attached image.

A is the location of airport.

First it travels for 210 miles southeast from A to B and then 210 miles south from B to C.

`\angle ABC = 135^\circ`

To find:

Side AC = ?

Solution:

As we can see, the
`\triangle ABC` is an isosceles triangle with sides AB = BC = 210 miles.

So, we can say that the angles opposite to the equal angles in a triangle are also equal.
`\angle A = \angle C`

And sum of all three angles of a triangle is equal to
`180^\circ`.

`\angle A+\angle B+\angle C = 180^\circ\\\Rightarrow \angle A+135^\circ+\angle A = 180^\circ\\\Rightarrow \angle A = (1)/(2) * 45^\circ\\\Rightarrow \angle A =22.5^\circ`

Now, we can use Sine Rule:

`(a)/(sinA) = (b)/(sinB)`

a, b are the sides opposite to the angles
`\angle A and \angle B` respectively.

`(210)/(sin22.5^\circ) = (b)/(sin135^\circ)\\\Rightarrow (210)/(sin22.5^\circ) = (b)/(cos45^\circ)\\\Rightarrow b = 210* (1)/(\sqrt2 * 0.3826)\\\Rightarrow b = 210* (1)/(0.54)\\\Rightarrow b \approx 388\ miles`

So, the answer is:

The plane is 388 miles far from the airport.