Answer:
about 54% of the drive
Explanation:
You want to know the fraction of the line segment between 75 miles north of a city and 85 miles east of the city that lies within 64 miles of a radio transmitter at the city center.
Parametric line
We can write a parametric equation for the line segment between points (0, 75) and (85, 0) as ...
L = (0, 75) +t((85, 0) -(0, 75)) = (0+85t, 75 -75t)
Circle
Using these values of (x, y) in the equation for the circle of radius 64 centered at the origin, we have ...
x² +y² = 64²
(85t)² +(75 -75t)² = 4096
12850t² -11250t +1529 = 0 . . . . . . write in standard form
Fraction
Remembering the quadratic formula, we know the two solution values for t will differ by ...
√(b² -4ac)/a
√(11250² -4(12850)(1529))/12850 = √47971900/12850 ≈ 0.539002
You will pick up a signal for about 53.9% of the drive.
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Additional comment
Effectively, we have solved the circle equation using the relation between x and y that is defined by the line x/85 +y/75 = 1.
Working the problem in the most straightforward way, we would determine the distance traveled between the point 75 miles north and the point 85 miles east. Then we would use the circle equation to find the points on the circle where it crossed the travel path. Using the distance formula again would give the distance we could hear the radio station. Then the desired fraction would be the ratio of that distance to the total travel distance.
We find the solution method above to be a lot less work.