Answer:
To find the height of the plane, we need to use trigonometry. Let's call the height of the plane "h". We can use the given angle of 57° and the opposite side (height) to the angle to find the adjacent side (distance traveled east) using the tangent function:
tan(57°) = h / distance traveled east
We can rearrange this equation to solve for h:
h = distance traveled east x tan(57°)
To find the distance traveled east, we need to use the given distance of 322 miles and the direction traveled. Since the plane is traveling at a 57° angle northeast, we can split this into two right triangles, one facing northeast and the other facing southeast, as shown below:
N
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|\ 57°
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The distance traveled east is the adjacent side of the southeast-facing right triangle, which can be found using the cosine function:
cos(57°) = distance traveled east / 322
We can rearrange this equation to solve for the distance traveled east:
distance traveled east = 322 x cos(57°)
Now we can plug in this value for the distance traveled east into the equation for the height of the plane:
h = distance traveled east x tan(57°)
h = (322 x cos(57°)) x tan(57°)
Using a calculator, we can evaluate this expression to find:
h ≈ 389.4 miles
Therefore, the height of the plane to the nearest mile is 389 miles.