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A small radio transmitter broadcasts in a 64 mile radius. If you drive along a straight line from a city 75 miles north of the transmitter to a second city 85 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

1 Answer

1 vote

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.

A small radio transmitter broadcasts in a 64 mile radius. If you drive along a straight-example-1
User Sachem
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