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 .
The given equation is x 2 + y 2 = z .
Equate the constant terms: r 2 = z .
Solve for r : r = z $. The radius is the square root of the constant term.
z
Explanation
Analyze the equation of a circle Let's analyze the equation of a circle centered at the origin, which is given by x 2 + y 2 = r 2 , where r represents the radius of the circle. In our case, we have the equation x 2 + y 2 = z . We need to find the relationship between z and the radius r .
Compare with the standard equation Comparing the given equation x 2 + y 2 = z with the standard equation x 2 + y 2 = r 2 , we can see that z corresponds to r 2 . Therefore, we have the equation r 2 = z .
Solve for the radius To find the radius r , we need to take the square root of both sides of the equation r 2 = z . This gives us r = z . T h ere f ore , t h er a d i u so f t h ec i rc l e i s t h es q u a reroo t o f t h eco n s t an tt er m z$.
Examples
Imagine you're designing a circular garden and you know the area it will cover can be represented by the equation x 2 + y 2 = 25 . To determine how much fencing you need (which depends on the radius), you take the square root of 25, which gives you a radius of 5 units. This tells you the garden will extend 5 units in all directions from the center, helping you plan your fence accordingly. This simple calculation ensures your circular garden fits perfectly within your designated space.
