A small radio transmitter

Question

A small radio transmitter broadcasts in a 22-mile radius. If you drive along a straight line from a city 27 miles north of the transmitter to a second city 28 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

x^2 + y^2 = 22^2 is a circle with radius 22, center at (0,0)

let y = mx + b be a line through the starting point (0,27) to the finish point (28,0)

slope = m = -27/28 b=27 = y intercept

y=-27x/28 + 27 is the line of travel

find where it intersects the circle, find 2 intersection points and calculate the distance between them

x^2 + (-27x/28 +27)^2 = 22^2

(729/784 + 1)x^2 - 2(27)(27)/28 + 28^2 -22^2 = 0

use the quadratic formula

or carefully graph the line and circle

If this problem were slightly different (and maybe it is if there's a typo) with the finish point at 27 miles east of the transmitter, the numbers simplify

x = + or minus sqr(22^2 - (-x+27)^2)

x^2 -27 +126=0

(x-21)(x-6) = 0

the points of intersection are (6,21) and (21,6)

they're a distance of 15sqr15 apart or about 21.21 miles

which means if you really did want to use 28 miles east, then the distance is slightly less than 21.21

probably about 21 miles