- The standard equation of a circle centered at the origin is \(x^2 + y^2 = r^2\), where \(r\) is the radius.
- Comparing this to the given equation \(x^2 + y^2 = z\), we see that \(z = r^2\).
- Solving for the radius, we find \(r = \sqrt{z}\).
- Thus, the correct explanation is that the radius is the square root of the constant term, which matches option C.
- Final Answer: \(\boxed{C}\)
Explanation
- Review the standard form of a circle equation.
The standard form of a circle's equation with a center at the origin \((0, 0)\) and a radius \(r\) is given by the formula \(x^2 + y^2 = r^2\). This equation is derived from the distance formula, where the distance from any point \((x, y)\) on the circle to the center \((0, 0)\) is constant and equal to \(r\). Squaring both sides of \(\sqrt{x^2 + y^2} = r\) yields \(x^2 + y^2 = r^2\).
- Compare the given equation with the standard form.
We are given the equation of a circle in the form \(x^2 + y^2 = z\). By comparing this to the standard form \(x^2 + y^2 = r^2\), we can identify that the constant term \(z\) corresponds to the square of the radius, meaning \(r^2 = z\).
- Determine the radius.
To find the radius \(r\), we solve the equation \(r^2 = z\) for \(r\). Taking the square root of both sides, we obtain \(r = \sqrt{z}\). This shows that the radius of the circle is the square root of the constant term \(z\). Therefore, option C is the correct explanation.
Examples
Suppose you are designing a circular fountain in a park and want to describe its placement on a coordinate grid. If you determine the area of the fountain is proportional to a value \(z\) such that its boundary equation is \(x^2 + y^2 = 25\), you now know that the radius of your fountain is \(\sqrt{25} = 5\) units. This helps in accurately marking the boundaries for construction and ensuring the fountain fits perfectly in the designated space.