Question

the wheel in the adjacent figure has a radius of 69 feet. the clearance between the wheel and the ground is 15 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.

Explanation:

Step1: Find the center of the circle

The radius of the wheel \(r = 69\) feet and the clearance between the wheel and the ground is \(15\) feet. The \(y\) - coordinate of the center of the circle \(h=69 + 15=84\). Since the origin is directly below the center of the wheel, the \(x\) - coordinate of the center \(k = 0\). So the center of the circle is \((h,k)=(0,84)\).

Step2: Use the standard form of the equation of a circle

The standard form of the equation of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\). Substitute \(h = 0\), \(k = 84\) and \(r = 69\) into the formula.
We get \((x-0)^{2}+(y - 84)^{2}=69^{2}\)

Answer:

\(x^{2}+(y - 84)^{2}=4761\)