displacement of a traveler

The figure above shows the route of Diane's trip from her house to her job. Diane travels 16 miles from A to B, 15 miles from B to C, and 4 miles from C to D. If she were able to travel from A to D directly, how much shorter, in miles, would the trip be?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 15

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Answer:

Choice C
Level of Difficulty E M H

The diagram shows the path that Diane travels, but then asks how much shorter the trip would be if she went directly from A to D. This distance is actually called the displacement, and is an important concept in physics.

The distance she traveled on the path she chose was 16 + 15 + 4 = 35 miles. Now let's get the distance from A to D. Then, we can take the difference between the two.

The key to getting the distance from A to D is to form a large rectangle that contains the entire diagram, with the lower left corner of the rectangle at A, and the upper right corner at D. Now you will notice that the direct path we are interested in is the diagonal of the rectangle. This diagonal is also the hypotenuse of a right triangle whose legs have length 20 (16+4) and 15. We can apply Pythagorean's Theorem to find the length of the hypotenuse:

C2 = A2 + B2

C2 = 202 + 152

C2 = 400 + 225

C2 = 625

C = 25

Now we can answer the question "If she were able to travel from A to D directly, how much shorter, in miles, would the trip be?" All we need to do is take the difference between the path she actually walked (35 miles) and the straight-line distance between A and D (25 miles) which gives 10 miles, or choice C.

You might also have noticed that the legs of a right triangle with lengths 15 (3X5) and 20 (4X5) must be the legs of a 3-4-5 right triangle, so the hypotenuse must be 25 (5X5).

If you are interested in learning more about displacement, visit The Physics Classroom.

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