Math word problem

Question

A small radio transmitter broadcasts in a 63 mile radius. If you drive along a straight line from a city 77 miles north of the transmitter to a second city 82 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Answer 1

Here are some hints, give this a try.

What is the equation of a circle with center at (0,0) and radius 63? The points in and on the circle are where the transmitter is able to reach.

The point (0,77) represents the point directly north of the transmitter.

The point (82,0) the point directly east of the transmitter.

The segment joining those points is the path taken by you (the driver).

Can you find the equation of the line passing through those points?

Can you find the points of intersection of that line and the equation of the circle?

Once you have the points of intersection, use the distance formula to find the distance between those points and you will have the part of the total distance that the driver picks up a signal.

You might be expected to calculate the % that distance is as part of the total drive.

Answer 2

This problem is a difficult problem in that it involves numbers that are large and not so nice to deal with. It also involves solving a non-linear system of equations, namely the intersection points of a line and a circle. The way that I solved this was to find the intersection points, subtract the distance between them, and then use the Pythagorean theorem to find the total distance of the drive. Doug is spot on with the process here.

You have the following system:

x²+y² = 63²

y =(-77/82)x + 77 ⇒ x = (y - 77) * -82/77

Substitute the equation of the line in for "x" in the circle equation and solve for the y values which would be the distance of the intersection points. I'll leave the algebra to you.

From there you would get y = (517748 ± 539*sqrt(211289))/12653 = 60.5, 21.338

Now, use Pythagorean theorem to find the distance travelled:

c²= 77² + 82² ⇒ c= 112.48 miles

Subtract 60.5 miles - 21.338 miles = 39.162 miles (# of miles driver is within range of transmitter)

To find the % of the trip that the driver could pick up the signal from the transmitter, set up an equation:

112.48 * X = 39.162 ⇒ X= 0.348 ≅ 35% of the drive.

So, 35% of the time is the answer.