Precalculus Question

Question

A small radio transmitter broadcasts in a 60 mile radius. If you drive along a straight line from a city 72 miles north of the transmitter to a second city 70 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter? Give your answer rounded to the nearest tenth of a mile.

_______ miles

Answer 1

Let x²+y²=60² represent the circular region over which the transmitter broadcasts (with transmitter at the origin).

Let point A = (0,72) be the point north of the transmitter.

Let point B = (70,0) be the point east of the transmitter.

Determine the equation of the line passing through A and B.

Find the points of intersection of the equation of the line and the circle.

Call those points P and Q.

Determine the distance between P and Q using the distance formula.

The calculations will be messy.

Here is a Desmos graph that does the calculations for you. Note the technique that is used to find the points of intersection using the Desmos ~ operator.

Hopefully you will try the suggestions for finding the answer before looking at the Desmos graph?

desmos.com/calculator/4ubfzrpy0r