To find the circumference of a circle given by the equation [tex]\( x^2 + y^2 = 16 \)[/tex], let's follow these steps:
1. Identify the equation's format:
The given equation [tex]\( x^2 + y^2 = 16 \)[/tex] is in the standard form of a circle equation, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], where [tex]\((h, k)\)[/tex] is the center and [tex]\(r\)[/tex] is the radius.
2. Determine the center:
Since there are no [tex]\(h\)[/tex] or [tex]\(k\)[/tex] terms modified in the expression (i.e., it is in the form [tex]\(x^2 + y^2\)[/tex] with no shifts), the center [tex]\((h, k)\)[/tex] is at [tex]\((0, 0)\)[/tex].
[tex]\[
\text{Center} = (0, 0)
\][/tex]
3. Find the radius:
The given equation [tex]\( x^2 + y^2 = 16 \)[/tex] implies that [tex]\( r^2 = 16 \)[/tex]. To find the radius [tex]\( r \)[/tex], take the square root of both sides.
[tex]\[
r = \sqrt{16} = 4
\][/tex]
4. Write the circumference formula:
The circumference [tex]\( C \)[/tex] of a circle is calculated using the formula [tex]\( C = 2\pi r \)[/tex].
5. Substitute the radius into the formula:
Using the radius [tex]\( r = 4 \)[/tex], we substitute into the circumference formula.
[tex]\[
C = 2\pi \times 4 = 8\pi
\][/tex]
Therefore, the center of the circle is [tex]\((0, 0)\)[/tex], the radius is [tex]\(4\)[/tex], and the circumference of the circle is [tex]\( 8\pi \)[/tex].
