To determine the radius of the circle given by the equation [tex]\( x^2 + y^2 = 64 \)[/tex], follow these steps:
1. Identify the general form of the equation of a circle:
The standard form of a circle's equation is [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. Compare the given equation with the standard form:
The given equation is [tex]\( x^2 + y^2 = 64 \)[/tex]. By comparing it with the standard form, you can see that:
- The center [tex]\((h, k)\)[/tex] is [tex]\((0, 0)\)[/tex] because there are no terms offsetting [tex]\( x \)[/tex] or [tex]\( y \)[/tex] (i.e., [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are not shifted by any amount).
- The right side of the equation [tex]\( 64 \)[/tex] represents [tex]\( r^2 \)[/tex].
3. Calculate the radius [tex]\( r \)[/tex]:
- The equation given is in the form [tex]\( r^2 = 64 \)[/tex].
- To find the radius [tex]\( r \)[/tex], take the square root of both sides of the equation:
[tex]\[
r = \sqrt{64}
\][/tex]
4. Find the numerical value:
- The square root of 64 is 8.
5. Conclusion:
Therefore, the radius of the circle [tex]\( x^2 + y^2 = 64 \)[/tex] is [tex]\( \boxed{8.0} \)[/tex].
