The equation (x + 9)² + (y - 4)² = 81 models the position and range of the source of a radio signal. Describe the position of the source and range of the signals.
The position of the source is (-9, 4), as this is the center of the circle. The range of the signal is 9 units, which is the square root of 81 (the radius of the circle). This means that any point within 9 units of the source will receive the radio signal.
Could you show how you got that answer?
Yes, I can show you step-by-step how I got that answer.
We start with the given equation:
(x + 9)² + (y - 4)² = 81
We can recognize that this is the equation of a circle, with center (-9,4) and radius 9. To see this, we can rewrite the equation in standard form:
(x - (-9))² + (y - 4)² = 9²
Simplifying, we get:
(x + 9)² + (y - 4)² = 81
So the center of the circle is (-9,4) and the radius is 9.
The position of the radio source is the center of the circle (-9, 4).
The range of the signals is the distance from any point on the circle to the center (-9, 4), which is the radius of the circle, or 9 units. This means that any point within 9 units of the source will receive the radio signal.
Can you solve this equation using the same method you did with the previous one
(x - 7)² + (y + 2)² = 64
Sure, I can solve this equation using the same method.
First, we can recognize that this is the equation of a circle, with center (7,-2) and radius 8. To see this, we can rewrite the equation in standard form:
(x - 7)² + (y + 2)² = 8²
So the center of the circle is (7,-2) and the radius is 8.
The position of the radio source is the center of the circle (7, -2).
The range of the signals is the distance from any point on the circle to the center (7, -2), which is the radius of the circle, or 8 units. This means that any point within 8 units of the source will receive the radio signal.
To describe the position and range of the source of a radio signal using the given equation (x + 9)² + (y - 4)² = 81, we can determine the center and radius of the equation.
Step 1: Compare the given equation with the standard equation of a circle, (x - h)² + (y - k)² = r². Here, the standard equation represents a circle with center (h, k) and radius r.
Comparing the given equation (x + 9)² + (y - 4)² = 81 with the standard equation, we can see that:
h = -9, k = 4, and r = √81 = 9.
Step 2: Interpret the results.
- The center of the circle is (-9, 4), which represents the position of the radio signal source.
- The radius of the circle is 9 units, which represents the range of the radio signals.
Therefore, the position of the radio signal source is at (-9, 4) and the range of the signals extends up to a distance of 9 units from the source.
To describe the position of the source and the range of the signals, we need to analyze the given equation:
(x + 9)² + (y - 4)² = 81.
This equation represents a circle in the coordinate plane.
The center of the circle is represented by the values (-9, 4), which corresponds to -9 for the x-coordinate and 4 for the y-coordinate. Therefore, the position of the source is at (-9, 4).
The radius of the circle can be found by taking the square root of 81, which equals 9. Thus, the range of the signals from the source is a distance of 9 units in all directions from the center.
In summary, the position of the source is at (-9, 4), and the range of the signals extends 9 units in all directions from the source.