Question

in real - world problems, the equation of a circle can be used to determine the between two points.
the value of r in the equation (x + 4)^2+(y - 1)^2 = 64 represents the of the circle.
a circles equation can help find the location of its.
the equation (x - 3)^2+(y + 2)^2 = 36 describes a circle with radius.
the radius of a circle is half the length of the.
the standard form of a circles equation is derived from the formula.
solving real - world problems using the equation of a circle often involves finding the of objects in space.

Explanation:

1. The distance formula between two points can be related to circle - equations in real - world problems where circles are used to model distances.
2. The standard form of a circle's equation is \((x - a)^2+(y - b)^2=r^2\), where \(r\) is the radius.
3. The center of a circle can be found from its equation in standard form \((x - a)^2+(y - b)^2=r^2\), with center \((a,b)\).
4. Comparing \((x - 3)^2+(y + 2)^2=36\) to \((x - a)^2+(y - b)^2=r^2\), since \(r^2 = 36\), then \(r = 6\).
5. The diameter \(d\) of a circle and radius \(r\) are related by \(d = 2r\), so \(r=\frac{d}{2}\).
6. The standard form of a circle's equation is derived from the distance formula.
7. Solving real - world problems with circle equations often involves finding the position of objects in space.

Answer:

1. distance
2. radius
3. center
4. 6
5. diameter
6. distance
7. position