Related Rates Circle Problem

See also: Related Rates Sphere Problem

The circumference of a circle is increasing at $11.6$ feet/second. When the radius is 8 feet, what rate (feet/sec) is the radius changing, and at what rate is the area changing?

We must first write the equation of the circumference in terms of time. $$C = 2\pi (8) + 11.6t$$ Just to check, we differentiate this equation. $\dfrac{dC}{dt} = 11.6$ and at $t = 0$, $r = 8$. So, our equation is correct. We can rewrite the above equation as $$2\pi r = 2\pi (8) + 11.6t$$ We divide by $2\pi$ and get $$r = 8 + 1.8462t$$ We differentiate, and find that $\dfrac{dr}{dt} = 1.8462$, 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 $$A = \pi r^2$$ We differentiate and use the chain rule: $$\dfrac{dA}{dt} = 2 \pi r \dfrac{dr}{dt}$$ At that instant, the radius = 8. The radius changes at 1.846 feet/second, so the rate at which the area changes is $$\dfrac{dA}{dt} = 2 \pi (8)(1.846) = \boxed{92.8} \text{ feet^2/sec}$$