A small radio transmitter broadcasts in a 36 mile radius. If you drive along a straight line from a city 41 miles north of the transmitter to a second city 46 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?


This problem involves finding how much of a straight-line path from one city to another lies within the coverage area of a radio transmitter, which broadcasts in a circular pattern with a 36-mile radius.

### Step 1: Visualizing the Problem
- The transmitter is located at the origin (0, 0).
- The first city is 41 miles directly north of the transmitter, so its coordinates are (0, 41).
- The second city is 46 miles directly east of the transmitter, so its coordinates are (46, 0).
- The radio signal covers a circle with a radius of 36 miles around the transmitter.

We need to find the part of the straight-line path between these two cities that lies within this circle.

### Step 2: Equation of the Circle
The equation for the circle with the transmitter at the center is:

\[
x^2 + y^2 = 36^2 = 1296
\]

### Step 3: Equation of the Line Between the Two Cities
The line connecting the two cities has endpoints at (0, 41) and (46, 0). To find the equation of the line, we use the slope formula:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 41}{46 - 0} = -\frac{41}{46}
\]

So the equation of the line in point-slope form is:

\[
y - 41 = -\frac{41}{46}(x - 0)
\]

or

\[
y = -\frac{41}{46}x + 41
\]

### Step 4: Finding the Intersection Points
To find the points where this line intersects the circle, substitute the equation of the line into the equation of the circle:

\[
x^2 + \left(-\frac{41}{46}x + 41\right)^2 = 1296
\]

This is a quadratic equation in \(x\), which we can solve to find the two intersection points. The difference between the \(x\)-coordinates of these points will give the length of the portion of the drive that lies within the signal range.

### Step 5: Solving the Quadratic Equation
Substituting \(y = -\frac{41}{46}x + 41\) into \(x^2 + y^2 = 1296\):

\[
x^2 + \left(-\frac{41}{46}x + 41\right)^2 = 1296
\]

This expands and simplifies to give a solvable quadratic equation. I'll now solve this for the exact answer. sympy import symbols, Eq, solve

# Define the variable
x = symbols('x')

# The equation of the line y = -41/46 * x + 41
y = (-41/46) * x + 41

# The equation of the circle x^2 + y^2 = 1296
circle_eq = Eq(x**2 + y**2, 1296)

# Solve the equation
solutions = solve(circle_eq, x)
solutionsThe line intersects the circle at \(x \approx 6.22\) miles and \(x \approx 34.51\) miles. These are the \(x\)-coordinates of the two points where the straight-line path crosses into the radio signal's range.

### Step 6: Calculate the Distance Within the Signal
The distance between these two \(x\)-coordinates is:

\[
\text{Distance} = 34.51 - 6.22 = 28.29 \, \text{miles}
\]

### Final Answer:
You will pick up the signal for approximately **28.29 miles** of the drive.

---

Would you like more details on any part of the solution, or do you have further questions?

Here are some related questions to explore:

1. How does the shape of the radio signal's coverage affect the area covered?
2. What other types of problems involve circles and lines?
3. How do we calculate the distance between two points of intersection?
4. What would happen if the radius of the signal was smaller or larger?
5. How can we apply this method to other geometric shapes?

**Tip**: If the problem involves more complex shapes or curves, breaking down the geometric relationships step-by-step (like we did here) makes it easier to find solutions!

A small radio transmitter broadcasts in a 36 mile radius. If you drive along a straight line from a city 41 miles north of the transmitter to a second city 46 miles east of the transmitter, during how much of the drive will you pick up a signal from the transmitter?

Learn how to calculate the distance along a straight-line drive that intersects a radio signal range with a radius of 36 miles. This problem combines geometric concepts such as circles and linear equations. Suitable for high school students in Grades 9-12.

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