Answer :
To describe the position of the source and the range of the signals modeled by the equation \((x+6)^2 + (y+4)^2 = 36\), we will interpret this equation in the context of the standard form of a circle equation \((x - h)^2 + (y - k)^2 = r^2\).
### Identifying the Center
The standard form of the equation of a circle is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
By comparing the given equation \((x+6)^2 + (y+4)^2 = 36\) with the standard form:
- We can see that the terms \((x + 6)\) and \((y + 4)\) correspond to \((x - (-6))\) and \((y - (-4))\) respectively, indicating that the center \((h, k)\) of the circle is at \((-6, -4)\).
### Identifying the Radius
The constant on the right-hand side of the equation, 36, corresponds to \(r^2\) in the standard form. To find the radius \(r\), we take the square root of 36:
[tex]\[ r = \sqrt{36} = 6 \][/tex]
### Conclusion
1. The position of the source of the radio signal, which corresponds to the center of the circle, is at \((-6, -4)\).
2. The range of the signal, which corresponds to the radius of the circle, is 6 units.
Hence, the position of the source of the radio signal is [tex]\((-6, -4)\)[/tex], and the range of the signals is 6 units.
### Identifying the Center
The standard form of the equation of a circle is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
By comparing the given equation \((x+6)^2 + (y+4)^2 = 36\) with the standard form:
- We can see that the terms \((x + 6)\) and \((y + 4)\) correspond to \((x - (-6))\) and \((y - (-4))\) respectively, indicating that the center \((h, k)\) of the circle is at \((-6, -4)\).
### Identifying the Radius
The constant on the right-hand side of the equation, 36, corresponds to \(r^2\) in the standard form. To find the radius \(r\), we take the square root of 36:
[tex]\[ r = \sqrt{36} = 6 \][/tex]
### Conclusion
1. The position of the source of the radio signal, which corresponds to the center of the circle, is at \((-6, -4)\).
2. The range of the signal, which corresponds to the radius of the circle, is 6 units.
Hence, the position of the source of the radio signal is [tex]\((-6, -4)\)[/tex], and the range of the signals is 6 units.
