Final answer:
The completed equation of the circle with a center at (5, 6) and radius of 3 is (x - 5)^2 + (y - 6)^2 = 9. The formula for the area of a circle is πr^2. The radius of a circle is central to many properties and applications, including geometry and physical kinematics.
Step-by-step explanation:
The student has provided the center of a circle and its radius, and needs to complete the standard formula of a circle based on this information. The standard form of the equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. Given the center (5, 6) and radius 3, the equation is therefore (x - 5)^2 + (y - 6)^2 = 9.
Reflecting on the mathematics of circles, we know that the area of a circle is given by the formula πr^2, where π is approximately 3.14159, and r is the radius. This formula applies no matter whether the circle fits neatly inside a square, as with a rectangle of side length 2r for a diameter of a circle, or in any other context.
In the case of circular motion, such as a spinning yo-yo, the tangential acceleration at the edge of the yo-yo is dependent on the radius and the angular acceleration. For a circle, understanding the radius is crucial not just for computing area and perimeter but also kinematic quantities such as tangential acceleration in physical applications.