The problem asks us to find how much of a straight-line journey between two cities, one 80 miles north and one 74 miles east of a radio transmitter, will be within the transmitter's 64-mile broadcast radius.

Step 1: Coordinates of key points

Let's define the location of the transmitter as the origin $(0, 0)$ on a coordinate plane.

• The first city (north of the transmitter) is at $(0, 80)$.
• The second city (east of the transmitter) is at $(74, 0)$.
• The transmitter has a broadcast radius of 64 miles, which forms a circle with the equation: $x^2 + y^2 = 64^2 = 4096$

Step 2: Equation of the line

The line of travel connects the two cities, so we need to find the equation of the line passing through $(0, 80)$ and $(74, 0)$.

The slope $m$ of the line is given by: $m = \frac{0 - 80}{74 - 0} = \frac{-80}{74} = -\frac{40}{37}$

Thus, the equation of the line in slope-intercept form is: $y = -\frac{40}{37}x + 80$

Step 3: Points of intersection with the circle

To find out where the line enters and exits the circle, we need to solve for the points where this line intersects the circle $x^2 + y^2 = 4096$. Substituting the equation of the line $y = -\frac{40}{37}x + 80$ into the circle equation:

$x^2 + \left( -\frac{40}{37}x + 80 \right)^2 = 4096$

Expanding the square: $x^2 + \left( \frac{1600}{1369}x^2 - \frac{6400}{37}x + 6400 \right) = 4096$ Multiplying through by 1369 to eliminate the fraction: $1369x^2 + 1600x^2 - 6400 \cdot 37 x + 6400 \cdot 1369 = 4096 \cdot 1369$ Solve for $x$ then use distance equationThe values of $x$ where the line intersects the circle are approximately $x = 16.90$ miles and $x = 62.86$ miles. These correspond to the points along the path where the signal begins and ends.

Step 4: Distance of the drive within the signal radius

To find how much of the drive is within the signal, we subtract the two $x$-coordinates: $62.86 - 16.90 = 45.96 \text{ miles}$

Thus, you will pick up the signal for approximately 46 miles of the drive.

Would you like more details on any step or have questions? Here are some related questions for further exploration:

1. How would the result change if the radius of the transmitter increased to 80 miles?
2. What is the total length of the drive between the two cities?
3. What would happen if the cities were located at different distances from the transmitter?
4. Can we generalize this method for any two cities at arbitrary distances?
5. How would the problem change if the travel path wasn't a straight line?

Tip: When solving problems involving geometric figures and lines, visualizing or sketching the problem often helps clarify the relationships between different points and distances.