The area of a circle is changing at a rate of $5 cm^2 s^{-1}$. Find the rate of change of the circumference at an instant when the radius is 2 cm.
Answer (1)
Write the formulas for the area A = π r 2 and circumference C = 2 π r of a circle.
Differentiate both formulas with respect to time t , obtaining d t d A = 2 π r d t d r and d t d C = 2 π d t d r .
Use the given d t d A = 5 and r = 2 to find d t d r = 4 π 5 .
Substitute d t d r into the expression for d t d C to find the rate of change of the circumference: d t d C = 2 5 .
Explanation
Problem Setup We are given that the area of a circle is changing at a rate of 5 c m 2 / s , which means d t d A = 5 . We want to find the rate of change of the circumference, d t d C , at the instant when the radius is 2 cm, i.e., r = 2 .
Area and Circumference Formulas The formula for the area of a circle is A = π r 2 . The formula for the circumference of a circle is C = 2 π r .
Differentiating the Area Formula Differentiating the area formula with respect to time t , we get d t d A = d t d ( π r 2 ) = 2 π r d t d r .
Solving for dr/dt We are given that d t d A = 5 and r = 2 . Substituting these values into the equation from the previous step, we have 5 = 2 π ( 2 ) d t d r , which simplifies to 5 = 4 π d t d r . Solving for d t d r , we get d t d r = 4 π 5 .
Differentiating the Circumference Formula Now, we differentiate the circumference formula with respect to time t : d t d C = d t d ( 2 π r ) = 2 π d t d r .
Substituting dr/dt Substituting the value of d t d r we found earlier into this equation, we get d t d C = 2 π ( 4 π 5 ) .
Final Calculation Simplifying the expression, we find d t d C = 4 π 10 π = 2 5 . Therefore, the rate of change of the circumference is 2 5 cm/s.
Examples
Imagine you're designing a circular fountain. Knowing how the circumference changes with the area helps you manage the water flow and the fountain's visual impact as the water surface expands or contracts. This calculation ensures the fountain's design remains balanced and aesthetically pleasing, demonstrating a practical application of related rates in engineering and design.
