A small radio transmitter broadcasts in a 23 mile radius. If you drive along a straight line from a city 31 miles north of the transmitter to a second city 32 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?
+118608
The equation of the line AB is \(y=\frac{-31}{32}x+31\)
The equation of the circle is \(x^2+y^2=23^2\)
I want to find the intersection.
So I need to solve
\(x^2+(\frac{-31}{32}x+31)^2=23^2\)
You can solve this as a standard quadratic if you want to but it is pretty awful so I used the wolframalpha calculator and go the the approx answers of x=11.351 and x=19.635
So between these values of x, the signal will be available 19.635-11.351 = 8.284
8.284/32 = 0.2588
So the signal will be available for approximately 25.88% of the drive.
calculation:
https://www.wolframalpha.com/input/?i=%28-31%2F32*x%2B31%29%5E2%2Bx%5E2%3D23%5E2
