Circle question-Precalculus

Question

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?

Answer 1

x^2 + y^2 = 23^2

is the equation of a circle which gives the outer limits of the radar. It's centered at the origin, (0,0)

31 north and 32 east are two points (0,31), (32,0) with a line through them. The equation of the line has slope -31/32 and y intercept = 31. The line is y=-31x/32 + 31.

find where the line intersects the circle. You should get two points. then use the distance formula to find the distance between them.

closest you'd come to the radar is when you're at the midpoint of the line or (16, 15.5)

16^2 + 15.5^2 = 256 + 240.25 = 496.25 < 23^2 = 529

y=-31x/32 + 31

15.5 = -31(16)/32 + 31 = 15.5

try the points that divide the line into a 1:3 ratio

-31(8)/32 +31 = 23.25 miles out at (0,8) and at (8,0)

which is close to 23 miles. (0,8) to (8,0) is at a distance of d, where d^2 = 2(8)^2 and d = 8sqr2 =about 8(1.1414) = about 11.314 miles within the radar's reach.

Since 23.25>23, maybe add a little to 11.314 to get an estimate of 11.4 miles?

After you do the tedious calculations, you should get something more than 11.3 miles

over 3/4 of the trip you're within the radar the entire trip is slightly over 44.55 miles

d^2 = 31^2 + 32^2, d=sqr1985 = 44.55333882 miles

Answer 2

The total distance between the cities is √(31²+32²) ≈ 44.55 miles

You need to find the intersection points between x²+y²=23² and y = -31x/32 +31.

(x₁,y₁) and (x₂,y₂) and calculate the distance between those two points. Then take the ratio

of that distance to the 44.55 mile distance between the two cities and multiply by 100 to get the percentage

of the time that you pick up the signal.