Related Rates Circle Problem — Mathwizurd

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Related Rates Circle Problem

MathJax TeX Test Page 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=2π(8)+11.6t Just to check, we differentiate this equation. dCdt=11.6 and at t=0, r=8. So, our equation is correct. We can rewrite the above equation as 2πr=2π(8)+11.6t We divide by 2π and get r=8+1.8462t We differentiate, and find that drdt=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=πr2 We differentiate and use the chain rule: dAdt=2πrdrdt At that instant, the radius = 8. The radius changes at 1.846 feet/second, so the rate at which the area changes is dAdt=2π(8)(1.846)=92.8 feet^2/sec
David Witten

Related Rates Sphere Problem

Proving limits of square roots