SOLUTION: A school provides bus service only to students who live a distance greater than 2 miles away from the school. On a Coordinate Plane, the school is located at the origin, and Michea

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Question 1183832: A school provides bus service only to students who live a distance greater than 2 miles away from the school. On a Coordinate Plane, the school is located at the origin, and Micheal lives at the closest point to the school on Maple Street, which can be represented by the line y=5x-4. If each unit on the Coordinate plane is 1 mile, does Micheal live far enough from school for bus service?
Found 2 solutions by greenestamps, Boreal:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


When you digest the given information, you will find that the problem is asking whether the closest point on the line y=5x-4 is more than 2 units from the origin.

The the distance from a point (p,q) to the line Ax+By+C=0 (i.e., the shortest distance from (p,q) to any point on the line) is given by this formula:

abs%28%28Ap%2BBq%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29%29

Changing the form of the equation of the line to the required form, we get 5x-y-4=0. Then plugging the appropriate numbers in the formula gives the distance from the origin to Michael's house as

abs%28%28%285%2A0%29%2B%281%2A0%29%2B4%29%2Fsqrt%285%5E2%2B1%5E2%29%29+=+4%2Fsqrt%2826%29

That is less than 1 mile...

ANSWER: No, Michael does not live far enough away from his school to qualify for getting bus service.

Even a rough sketch of the line with equation y=5x-4 shows that the shortest distance from the origin to the line is far less than 2; in fact it is clearly less than 1:

graph%28400%2C400%2C-2%2C4%2C-10%2C4%2C5x-4%29

Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
Graph this:
graph%28300%2C300%2C-3%2C3%2C-3%2C3%2Csqrt%28-x%5E2%2B4%29%2C-sqrt%28-x%5E2%2B4%29%2C5x-4%29
It should be clear from this that he would not qualify.
Finding the closest point is more work but can be done by using y=5x-4 and x^2+y^2=4
you can square y and have x^2+25x^2-40x+16=4
that will give x=0.41 and 1.13 where Maple St. intersects the 2 mile radius.
From the length of the chord (need y values) one may determine the length of the chord. Half that is the center and the closest point to the origin. Since the the length of the chord is known (one leg) and the hypotenuse is the radius, the closest point may be found, since this is a right triangle.