transmitter broadcast

Question

A small radio transmitter broadcasts in a 39 mile radius. If you drive along a straight line from a city 49 miles north of the transmitter to a second city 50 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

find 2 points where the circle intersects the line, the calculate the distance between them

the circle is x^2 +y^2 = 39^2, with center at the origin, radius=39

y^2 = 39^2- x^2

the line is doward sloping with x and y intercepts = (50,0) and (0,49)

slope = -49/50 = -.98

equation of the line is

y=mx+b

y=-49x/50 + 49

square that and set it = to the circle's value of y^2

or using a graphing calculator

the points the line intersects the circle are about (12.2,37) and(36.8,12.9)

use the distance formula

d= sqr((37-12.9)^2 +(36.8-12.2)^2)

= sqr(24.1^2 +24.6^2)

= about 34.44 miles

total distance traveled between the 2 cities is about 70 miles = sqr(49^2+50^2)=about 50sqr2=50(1.4)=70

about half the trip is within distance of the radio transmissions