See also: Related Rates Sphere Problem
We must first write the equation of the circumference in terms of time. Just to check, we differentiate this equation. and at , . So, our equation is correct. We can rewrite the above equation as We divide by and get We differentiate, and find that , meaning the radius changes at 1.8462 feet/second. Now, in order to find the rate of change of the area, we can first write an equation for area We differentiate and use the chain rule: At that instant, the radius = 8. The radius changes at 1.846 feet/second, so the rate at which the area changes is
David Witten