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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?

 Sep 28, 2021
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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

 

 Sep 28, 2021

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