First, let's recognize the equation of the circle provided:
[tex]\[
(x+3)^2 + (y-4)^2 = 64
\][/tex]
This equation is in the standard form of a circle's equation, which is:
[tex]\[
(x-h)^2 + (y-k)^2 = r^2
\][/tex]
Here, [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] represents the radius.
### Step-by-Step Solution:
1. Compare the given equation [tex]\((x+3)^2 + (y-4)^2 = 64\)[/tex] with the standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex].
2. From the given equation, we can extract the values for [tex]\(h\)[/tex] and [tex]\(k\)[/tex]:
- [tex]\((x+3)^2\)[/tex] implies [tex]\(h = -3\)[/tex]
- [tex]\((y-4)^2\)[/tex] implies [tex]\(k = 4\)[/tex]
Therefore, the center of the circle is [tex]\((-3, 4)\)[/tex].
3. The right-hand side of the equation [tex]\((x+3)^2 + (y-4)^2 = 64\)[/tex] is [tex]\(r^2\)[/tex], which is the square of the radius. From this, we determine:
- [tex]\(r^2 = 64\)[/tex]
- To find the radius [tex]\(r\)[/tex], we take the square root of 64:
[tex]\[
r = \sqrt{64} = 8
\][/tex]
Therefore, the center of the circle is [tex]\((-3, 4)\)[/tex] and the radius of the circle is [tex]\(8\)[/tex] meters.
### Conclusion:
The correct combination of the center and radius for the described circular walking path is:
[tex]\[
\text{Center at } (-3, 4) ; r = 8
\][/tex]
So, the correct option is:
Center at [tex]\((-3,4) ; r=8\)[/tex]
