Question

A small radio transmitter broadcasts in a 48 mile radius. if you drive along a straight line from a city 61 miles north of the transmitter to a second city 62 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

40.7 miles.
   For this problem, we want to know the length of the chord created by the line and the circle. So let's first create the equations needed.

   The slope intercept equation for a line is:
 y = ax + b
 the value for a will be the the difference in y divided by the difference in x. We're going from y=61 to y=0 for a chance of -61 and from x=0 to x=62 for a change of 62. So the value of a is
-61/62, giving us the formula
 y = -(61/62)x + b
 Substituting x = 0, we can calculate b
 61 = -(61/62)0 + b
 61 = b
   So the equation for the line is: y = -(61/62)x + 61

   Now for the equation for the circle. Since the circle is centered at the origin, the equation is:
 x^2 + y^2 = 48^2

   Now we to calculate the intersections.
 y = -(61/62)x + 61
 x^2 + y^2 = 48^2
 x^2 + (-(61/62)x + 61)^2 = 48^2
 x^2 + (3721/3844)x^2 - (3721/31)x + 3721 = 2304
 (7565/3844)x^2 - (3721/31)x + 3721 = 2304
 (7565/3844)x^2 - (3721/31)x + 1417 = 0
 1.968002081x^2 - 120.0322581x + 1417 = 0

   And we have a rather ugly quadratic equation which we can solve using the quadratic formula, giving the solutions x = 16.00512574 and x = 44.98681081

   Now we need to calculate the y values for those 2 x values.
 y = -(61/62)x + 61
 y = -(61/62)16.00512574 + 61 y = 45.25302145
   y = -(61/62)x + 61
 y = -(61/62)44.98681081 + 61
 y = 16.73878292

   So the 2 endpoints are (16.00512574, 45.25302145) and (44.98681081, 16.73878292)
   The distance between those points can be calculated using the Pythagorean theorem.
 sqrt((16.00512574 - 44.98681081)^2 + (45.25302145 - 16.73878292)^2) = sqrt(-28.98168506^2 + 28.51423853^2) =
sqrt(839.938069 + 813.0617988) =
sqrt(1652.999868) = 40.65710107

   And finally, we have the solution of 40.7 miles.