To determine how much of the drive you'll receive the signal, we must calculate the length of the line segment along which the car is within the 60-mile broadcast radius of the transmitter.

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Step 1: Represent the Problem in Coordinate Geometry

We place the transmitter at the origin $(0, 0)$. The cities are:

• The first city is 80 miles north, so it has coordinates $(0, 80)$.
• The second city is 73 miles east, so it has coordinates $(73, 0)$.

The line segment representing the drive is the line between $(0, 80)$ and $(73, 0)$.

The broadcast radius is a circle centered at $(0, 0)$ with a radius of 60 miles.

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Step 2: Equation of the Line

The equation of the line passing through $(0, 80)$ and $(73, 0)$ can be found using the slope-intercept form: $y = mx + b$

• The slope $m$ is: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 80}{73 - 0} = -\frac{80}{73}.$
• The $y$-intercept $b$ is 80 (since the line passes through $(0, 80)$).

Thus, the equation of the line is: $y = -\frac{80}{73}x + 80.$

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Step 3: Intersection with the Circle

The equation of the circle is: $x^2 + y^2 = 60^2 = 3600.$

Substitute $y = -\frac{80}{73}x + 80$ into the circle's equation: $x^2 + \left(-\frac{80}{73}x + 80\right)^2 = 3600.$

Expand the squared term: $\left(-\frac{80}{73}x + 80\right)^2 = \left(-\frac{80}{73}x\right)^2 + 2\left(-\frac{80}{73}x\right)(80) + 80^2,$ $\left(-\frac{80}{73}x + 80\right)^2 = \frac{6400}{5329}x^2 - \frac{12800}{73}x + 6400.$

Substitute this back: $x^2 + \frac{6400}{5329}x^2 - \frac{12800}{73}x + 6400 = 3600.$

Combine like terms: $\left(1 + \frac{6400}{5329}\right)x^2 - \frac{12800}{73}x + 6400 = 3600.$

Simplify: $\frac{5329 + 6400}{5329}x^2 - \frac{12800}{73}x + 6400 = 3600.$ $\frac{11729}{5329}x^2 - \frac{12800}{73}x + 6400 - 3600 = 0.$ $\frac{11729}{5329}x^2 - \frac{12800}{73}x + 2800 = 0.$

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Step 4: Solve for $x$

Multiply through by $5329 \cdot 73$ to eliminate fractions: $(11729)(73)x^2 - (12800)(5329)x + (2800)(5329)(73) = 0.$

Solve this quadratic equation for $x$, then calculate $y$ for the corresponding points. The length of the segment within the circle can be found by calculating the distance between the two intersection points.

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