Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

The signal from a radio station has a circular range of 50 miles. A second radio station, located 100 miles east and 80 miles north of the first station, has a range of 80 miles. Are there locations where signals can be received from both radio stations? Explain your answer.

Step-by-Step Solution

Answer

Yes, signals can be received from both stations as the circles intersect.

Step 1: Define the Center and Radius of Circles

Consider the first radio station as a circle centered at the origin (0,0) with a radius of 50 miles. The equation of this circle is \(x^2 + y^2 = 50^2\). The second radio station is located at (100, 80) with a range of 80 miles. Its equation is \((x - 100)^2 + (y - 80)^2 = 80^2\).

Step 2: Check for Intersection

To find if there are common areas, we need to determine if the circles intersect. For this, compute the distance between the centers of the circles, which is \( \sqrt{(100 - 0)^2 + (80 - 0)^2} = \sqrt{100^2 + 80^2} = 128 \).

Step 3: Compare Distance with Sum and Difference of Radii

The circles will intersect if the distance between centers (128) is less than or equal to the sum of the radii (50 + 80 = 130) but greater than the absolute difference of the radii (80 - 50 = 30). Since 30 < 128 < 130, the circles do intersect.

Step 4: Conclusion

Since the distance between the centers of the two circles falls within the range where the circles intersect, there are indeed locations where signals can be received from both radio stations.

Key Concepts

Circle Intersection

When we talk about circle intersection in geometry problems, we are looking to see where two circles overlap. This can be crucial for real-world applications, like determining where two radio station signals intersect. To find out if two circles intersect, follow these steps:

• Find the center and radius of each circle, which is often given in the form of equations. This helps to position the circles on a coordinate plane.
• Calculate the distance between the centers of the circles using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• Compare this distance to the sum and the difference of the radii of the two circles. If the distance is less than the sum of their radii and more than their absolute difference, the circles will intersect.

In our radio station example, since the distance between the centers is 128 miles and it falls between 30 and 130 miles, the two circles intersect. This indicates there are places where both radio signals can be received simultaneously.

Distance Between Points

Calculating the distance between two points on a plane is a fundamental concept in geometry. It is often used to solve problems involving the positioning of objects. The formula for calculating this is called the distance formula, which is derived from the Pythagorean theorem. This formula is:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• Here, \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points.
• The difference in the x-values is squared and added to the square of the difference in the y-values. The sum is then squared rooted to find the actual distance.

In our exercise, we used this formula to find that the distance between the two radio station centers is 128 miles. This distance helps us determine whether their coverage areas overlap.

Radio Station Range

A radio station's range can be imagined as an invisible circle spreading out from its location. This range is the maximum distance its signal reaches and can be represented by a circle equation. The real trick is when two stations' ranges overlap—finding these common areas can be solved using circle geometry.

• Each radio station can be represented mathematically as a circle, with its range being the radius.
• The intersection of these circles shows where both stations' signals can be received.
• These intersections are crucial for determining areas served by multiple stations, which can be important for maximizing coverage.

In our scenario, the first station's range is represented by a circle with a radius of 50 miles, while the second station has a bigger range, a circle with an 80-mile radius. Where these circles intersect, indicates regions where listeners can enjoy both signals.

Geometry Problems

Geometry problems involving circles, like our example, frequently appear in real-world scenarios and academic exercises. Solving these problems involves understanding equations of circles and how they interact.

• Circle equations, of the form \((x - h)^2 + (y - k)^2 = r^2\), describe circles on a coordinate plane where \((h, k)\) is the center and \(r\) is the radius.
• Solving geometry problems often involves finding intersections, distances, and areas related to these circles.
• The key is to visualize the problem, map out the given information, and apply mathematical concepts such as the distance formula and comparisons of radii sums and differences.

By understanding these concepts, students can tackle a variety of geometry problems, like determining overlapping coverage areas of radio stations, with ease. Such problems improve spatial reasoning and mathematical thinking, useful skills both inside and outside of academia.