Video: Resultant Motion and Force

A length of road stretches north for 10 km from the edge of a town to where it intersects an eastward road. A car is broken down on the eastward road, and the displacement of the edge of the town from the car has a magnitude of 24 km. How far east of the intersection is the car, to the nearest kilometer?

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Video Transcript

A length of road stretches north for 10 kilometers from the edge of a town to where it intersects an eastward road. A car is broken down on the eastward road, and the displacement of the edge of the town from the car has a magnitude of 24 kilometers. How far east of the intersection is the car, to the nearest kilometer?

We’ll label this distance 𝐷, and we’ll start off by drawing a diagram of this situation. We’re told that we have a town with a road from the eastern edge of the town that moves up north. This northward-pointing road intersects another road, which allows travel to the east. Along this east–west road, at some location, there is a car broken down.

We’re told that the distance between the broken down car and the town is 24 kilometers. And we’re also told that the stretch of road that goes to the north before it intersects the eastern road is 10 kilometers. Given this information, we want to solve for 𝐷, which is the distance from the broken down car to the northward-pointing road along the eastward road.

Since the two roads are perpendicular to one another, what we have is a right triangle with a hypotenuse of 24 kilometers and one leg of 10 kilometers. We can recall the Pythagorean theorem, which says that, for a right triangle with sides 𝐴 and 𝐵 and hypotenuse 𝐶, the length 𝐶 squared is equal to the length 𝐴 squared plus the length 𝐵 squared.

Applying this theorem to our scenario, we can say that 24 kilometers squared is equal to 10 kilometers squared plus 𝐷 squared, or that 𝐷 is equal to the square root of 24 kilometers squared minus 10 kilometers squared. Entering this expression on our calculator, we find, to the nearest kilometer, that 𝐷 is 22 kilometers. That’s the distance along the eastward road that the broken down car is from the northward-pointing road.

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