To determine the greatest distance a person could be from the lamp and still be detected by the motion detector, we need to analyze the given equation of the boundary.
The given equation is:
[tex]\[
(x + 16)^2 + (y - 13)^2 = 36
\][/tex]
This equation represents the boundary of a circle in its standard form:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
From the equation [tex]\((x + 16)^2 + (y - 13)^2 = 36\)[/tex], we can identify the following:
- The center of the circle, [tex]\((h, k)\)[/tex], is deduced by comparing it with the standard form. Thus, [tex]\(h = -16\)[/tex] and [tex]\(k = 13\)[/tex].
- The term on the right side of the equation, 36, represents [tex]\(r^2\)[/tex], which is the square of the radius.
To find the radius [tex]\(r\)[/tex], we take the square root of 36:
[tex]\[
r = \sqrt{36} = 6
\][/tex]
Therefore, the radius of the circle is 6 feet.
The greatest distance a person could be from the lamp and still be detected by the motion detector is equal to the radius of the circle.
Thus, the greatest distance is:
[tex]\[
\boxed{6 \text{ ft}}
\][/tex]
