Which explains how to find the radius of a circle whose equation is in the form $x^2+y^2=z$?
A. The radius is the constant term, $z$.
B. The radius is the constant term, $z$, divided by 2.
C. The radius is the square root of the constant term, $z$.
D. The radius is the square of the constant term, $z$.
Answer (1)
The equation of a circle centered at the origin is x 2 + y 2 = r 2 .
Compare the given equation x 2 + y 2 = z with the standard equation.
Equate the constant terms: r 2 = z .
Solve for r by taking the square root: r = z .
Explanation
Analyze the problem and data. The equation of a circle centered at the origin is given by x 2 + y 2 = r 2 , where r is the radius of the circle. We are given the equation x 2 + y 2 = z . Our goal is to find the radius of the circle in terms of z .
Compare with the standard equation. Comparing the given equation x 2 + y 2 = z with the standard equation x 2 + y 2 = r 2 , we can see that z corresponds to r 2 . Therefore, we have the relationship r 2 = z .
Solve for the radius. To find the radius r , we need to take the square root of both sides of the equation r 2 = z . This gives us r = z $.
State the final answer. Therefore, the radius of the circle is the square root of the constant term z .
Examples
Imagine you're designing a circular garden and you know the equation that defines its boundary is x 2 + y 2 = 25 . To determine how much fencing you need, you first need to find the radius. Since the radius is the square root of the constant term, in this case, it's 25 = 5 units. Knowing the radius, you can calculate the circumference (the amount of fencing needed) using the formula C = 2 π r , which would be 2 π ( 5 ) = 10 π units. This example shows how finding the radius from the equation of a circle is a practical step in real-world applications.
