Question

a small radio transmitter broadcasts in a 28 mile radius. if you drive along a straight line from a city 39 miles north of the transmitter to a second city 35 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter? miles

Explanation:

Step1: Find the distance between the two cities

Use the Pythagorean theorem \(d=\sqrt{39^{2}+35^{2}}\).

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Step2: Find the length of the chord

Let the radius of the circle \(r = 28\), and the distance from the center of the circle (transmitter) to the line (the two - city line) be \(h\). The area of the right - triangle formed by the two legs (\(39\) and \(35\)) can also be expressed as \(A=\frac{1}{2}\times d\times h=\frac{1}{2}\times39\times35\). So \(h=\frac{39\times35}{d}\approx\frac{39\times35}{52.4}\approx26.1\).
Then use the formula for the length of a chord \(L = 2\sqrt{r^{2}-h^{2}}\).

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Answer:

\(20.3\)