To find the equation of a circle centered at the origin (0, 0) with a given radius, we need to understand the standard form of the equation of a circle. The standard form for a circle centered at the origin is:
[tex]\[
x^2 + y^2 = r^2
\][/tex]
Here, [tex]\( r \)[/tex] represents the radius of the circle.
Given:
Radius ([tex]\( r \)[/tex]) = 15
Step-by-Step Solution:
1. We start by plugging the radius value into the standard equation of the circle.
[tex]\[
r = 15
\][/tex]
2. Substitute the radius into the standard form equation:
[tex]\[
x^2 + y^2 = (15)^2
\][/tex]
3. Calculate [tex]\( 15^2 \)[/tex] to find the value on the right-hand side of the equation:
[tex]\[
15^2 = 225
\][/tex]
4. So, the equation of the circle becomes:
[tex]\[
x^2 + y^2 = 225
\][/tex]
Looking at the provided options:
A. [tex]\( x^3 + y^3 = 225 \)[/tex] is incorrect because it involves cubic terms instead of quadratic terms.
B. [tex]\( x^2 + y^2 = 15 \)[/tex] is incorrect because the radius squared (225) should be on the right side of the equation, not just the radius.
C. [tex]\( x^3 + y^3 = 15 \)[/tex] is incorrect because it involves cubic terms and incorrectly places the radius.
D. [tex]\( x^2 + y^2 = 225 \)[/tex] is correct as it matches the standard form equation derived for a radius of 15.
Therefore, the correct answer is:
[tex]\[ \boxed{D \; x^2 + y^2 = 225} \][/tex]
