Question

the wheel in the adjacent figure has a radius of 65 feet. the clearance between the wheel and the ground is 13 feet. the rectangular coordinate system shown has its origin on the ground directly below the center of the wheel. use the coordinate system to write the equation of the circular wheel. the equation of the circular wheel is (simplify your answer. type your answer in standard form.)

Explanation:

Step1: Find the center of the circle

The origin is on the ground and the center of the wheel is \(65 + 13=78\) units above the \(x -\)axis (ground). Since there is no horizontal shift (assuming the wheel is centered in the coordinate system in the relevant direction), the center of the circle \((h,k)=(0,78)\)

Step2: Recall the standard form of the circle equation

The standard form of the equation of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center of the circle and \(r\) is the radius.

Step3: Substitute the values of \(h\), \(k\), and \(r\) into the equation

We know that \(h = 0\), \(k=78\), and \(r = 65\). Substituting these values into the standard - form equation \((x - 0)^{2}+(y - 78)^{2}=65^{2}\)

Answer:

\(x^{2}+(y - 78)^{2}=4225\)