Answer:
Step-by-step explanation
Suppose the transmitter is at the origin of a co-ordinate system. Then the range of the signal is the interior of a circle with the equation
x^2 + y^2 = 64^2.
The route the driver would take is from
City to the north at (0,71) as (x1,y1)
City to the east at (87,0) as (x2,y2), which is a line with the equation
y = mx+c where m is the slope
Slop m = (y2--y1)/(x2-x1) = (0-71)/(87-0) = -71/87
Intercept C = y - mx = 71 - (-71/87)(0) = 71
So the equation of line is y = (-71/87)x+71
Now we need to find the points of intersection of the circle and the line,
So substitute y = -(71/87)*x + 71 in the equation of circle
which is x^2 + y^2 = 64^2:
x^2 + (-(71/87)*x + 71)^2 = 64^2
x^2 + (71/87)^2 * x^2 - (2*(71/87)*71x) + 71^2 = 64^2
(1 + (71/87)^2)*x^2 - (10082/87)*x + (71^2 - 64^2) = 0
(12610/7569)*x^2 - (10082/87)*x + 945 = 0.
Use the quadratic formula: to find the roots of the equation
We get,
x = 9.437 to x = 59.957 miles.
So the signal will be picked up from x = 24.098 to x = 45.794 miles.