Which explains how to find the radius of a circle whose equation is in the form [tex]x^2+y^2=z[/tex]?
A. The radius is the constant term, [tex]z[/tex].
B. The radius is the constant term, [tex]z[/tex], divided by 2.
C. The radius is the square root of the constant term, [tex]z[/tex].
D. The radius is the squad of the constant term, [tex]z[/tex].
Answer (1)
Recognize the standard equation of a circle centered at the origin: x 2 + y 2 = r 2 .
Compare the given equation x 2 + y 2 = z with the standard equation, noting that z = r 2 .
Solve for the radius r by taking the square root of z : r = z .
Conclude that 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.
Compare with the given equation Now, let's compare this standard form with the given equation: x 2 + y 2 = z We can see that z corresponds to r 2 in the standard equation.
Solve for the radius To find the radius r , we need to take the square root of both sides of the equation z = r 2 :
r = z 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 area that the garden 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 would take the square root of 25. So, 25 = 5 . This tells you the radius of your garden is 5 units, which you can then use to calculate the circumference and determine the amount of fencing required. This concept is also fundamental in fields like astronomy, where calculating the radii of celestial bodies is crucial for understanding their physical properties and behavior.
