A small radio transmitter broadcasts in a 32 mile radius. If you drive along a straight line from a city 43 miles north of the transmitter to a second city 36 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?


We are asked to determine the portion of a straight drive, from a city located 43 miles north of a radio transmitter to a second city located 36 miles east of the transmitter, during which the radio signal can be picked up. The radio signal has a broadcast radius of 32 miles.

### Steps to Solve:
1. **Coordinate System Setup:**
   Let's place the transmitter at the origin of a coordinate plane, i.e., \((0, 0)\). The first city is 43 miles north, so its coordinates are \((0, 43)\). The second city is 36 miles east, so its coordinates are \((36, 0)\).

2. **Equation of the Line:**
   To describe the drive path, we need the equation of the straight line connecting the two cities. We can find the slope \(m\) of the line using the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 43}{36 - 0} = \frac{-43}{36}
   \]
   So, the equation of the line in point-slope form (using point \((0, 43)\)) is:
   \[
   y - 43 = \frac{-43}{36}(x - 0)
   \]
   Simplifying:
   \[
   y = -\frac{43}{36}x + 43
   \]

3. **Condition for Receiving Signal:**
   The transmitter broadcasts within a circle of radius 32 miles centered at the origin. The equation of this circle is:
   \[
   x^2 + y^2 = 32^2 = 1024
   \]
   We need to determine the points where the driving path intersects this circle, which will tell us the section of the path where the signal is available.

4. **Substitute the Line Equation into the Circle Equation:**
   Substitute \(y = -\frac{43}{36}x + 43\) into the circle equation:
   \[
   x^2 + \left(-\frac{43}{36}x + 43\right)^2 = 1024
   \]
   Expanding \(\left(-\frac{43}{36}x + 43\right)^2\):
   \[
   x^2 + \left(\frac{1849}{1296}x^2 - 2 \cdot \frac{43}{36} \cdot 43 \cdot x + 43^2\right) = 1024
   \]
   Simplifying further:
   \[
   x^2 + \frac{1849}{1296}x^2 - \frac{2 \cdot 43^2}{36}x + 1849 = 1024
   \]
   \[
   x^2 \left(1 + \frac{1849}{1296}\right) - \frac{2 \cdot 43^2}{36}x + 1849 = 1024
   \]
   After simplifying and solving for \(x\), we can find the points where the path intersects the circle.

5. **Calculate the Distance:**
   Once we have the \(x\)-coordinates of the intersection points, we can calculate the distance between them, which represents the portion of the drive during which the signal is picked up.

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A small radio transmitter broadcasts in a 32 mile radius. If you drive along a straight line from a city 43 miles north of the transmitter to a second city 36 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Learn how to determine the portion of a drive within a 32-mile radius of a radio transmitter using coordinate geometry. This problem covers the equation of a line, circle, and their intersection. Suitable for high school students (Grades 10-12).

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