Please help, my homework is due in a couple of days and I'm stuck. A plane is travelling 10.0 meters per second (this means the plane would travel ten meters in one second, twenty meters in two seconds, thirty meters in three seconds, and so on) north, but then a strong wind of about 15.0 meters per second begins to blow east. (a) Is the speed of the plane equal to, greater than, or less than 25.0 meters per second? (Do not make any calculations.) (b) Explain your reasoning for your answer above. (c) Is the direction of the plane north, east, or somewhere between? (d) [!] If you said somewhere between, write (as an algebraic inequality, if you can) for the angle at which the plane might be travelling once the wind begins to blow.

Explanation:

(a) Just as one-dimensional numbers add on a number line by putting them end-to-end, so two-dimensional numbers add on a coordinate plane the same way.

Here, we choose to let the positive y-axis represent North, and the positive x-axis, East. This is the way a map is conventionally oriented. The velocity of the plane is represented by a vector pointing north (up). Its length represents the magnitude of the velocity. Likewise, the wind is represented by a vector of length 15 pointing east (right). The sum of these is the hypotenuse of the triangle they form.

The magnitude of the sum can be found here using the Pythagorean theorem, but for the purpose of this question, you're not asked to find that.

Instead, you're asked to estimate whether it is more or less than 25 (m/s).

Your knowledge of the triangle inequality will tell you that the hypotenuse (resultant) must be shorter than the sum of the lengths of the sides of the triangle, hence must be less than 10+15 = 25.

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(b) The triangle inequality says the resultant is less than the sum of the other two sides of the triangle.

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(c) Since the wind is blowing the plane toward the east, but the plane is traveling toward the north, the resulting direction is somewhere between north and east.

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(d) "Somewhere between north and east" can be expressed as the inequality ...

  0° < angle < 90°