Question

(42) A school only provides bus service

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 Michael lives at the closest
point to the school on Maple Street,
which can be represented by the line
y = 2x – 4. If each unit on the coordinate
plane represents 1 mile, does Michael
live far enough from the school for bus
service?

Answer 1

~1.8 mile

Explanation:

Michael lives at the closest point to the school (the origin) on Maple Street, which can be represented by the line y = 2x – 4.

This means Michael's house will be the intersection point of line y1 (y = 2x - 4) and line y2 that is perpendicular to y1 and passes the origin.

Denote equation of y2 is y = ax + b,

with a is equal to negative reciprocal of 2 => a = -1/2

y2 pass the origin (0, 0) => b = 0

=> Equation of y2:

y = (-1/2)x

To find location of Michael's house, we get y1 = y2 or:

2x - 4 = (-1/2)x

<=> 4x - 8 = -x

<=> 5x = 8

<=> x = 8/5

=> y = (-1/2)x = (-1/2)(8/5) = -4/5

=> Location of Michael' house: (x, y) = (8/5, -4/5)

Distance from Michael's house to school is:

D = sqrt(x^2 + y^2) = sqrt[(8/5)^2 + (-4/5)^2) = ~1.8 (mile)