To address the communication range of a tower situated within a grid, consider the equation [tex]\(x^2 + y^2 = 64\)[/tex]. We'll analyze this equation step by step to determine the location of the tower (the center of the circle) and the radius of the signal coverage.
1. Understand the Equation:
The equation [tex]\(x^2 + y^2 = 64\)[/tex] represents a circle in a Cartesian coordinate system. This is a standard form of a circle's equation, which is generally written as:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the coordinates of the center of the circle, and [tex]\(r\)[/tex] the radius of the circle.
2. Identify the Center:
In our case, the equation provided is:
[tex]\[
x^2 + y^2 = 64
\][/tex]
Comparing this with the general form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we see that [tex]\(h = 0\)[/tex] and [tex]\(k = 0\)[/tex]. Therefore, the center of the circle is:
[tex]\[
(h, k) = (0, 0)
\][/tex]
3. Determine the Radius:
The term on the right-hand side of the equation [tex]\(x^2 + y^2 = 64\)[/tex] is [tex]\(64\)[/tex], which corresponds to [tex]\(r^2\)[/tex] in the general circle equation. To find [tex]\(r\)[/tex], the radius, we take the square root of 64:
[tex]\[
r = \sqrt{64} = 8
\][/tex]
4. Conclusion:
Based on these calculations, we can conclude that the communication tower is located at the center of the grid with coordinates [tex]\((0, 0)\)[/tex] and it sends signals within a circular area that has a radius of 8 units.
Summarizing, the center of the communication tower's signal coverage is at [tex]\((0, 0)\)[/tex] and the radius of this coverage is 8 units.
