Question

1. a circle has a radius of 8 inches which is changing. write the following formulas. circumference change of circumference area change of area

Explanation:

Step1: Write the formula for circumference

The formula for the circumference \(C\) of a circle is \(C = 2\pi r\).

Step2: Differentiate the circumference formula with respect to time \(t\)

Using the chain - rule \(\frac{dC}{dt}=\frac{d}{dt}(2\pi r)\). Since \(2\pi\) is a constant, \(\frac{dC}{dt}=2\pi\frac{dr}{dt}\).

Step3: Write the formula for area

The formula for the area \(A\) of a circle is \(A=\pi r^{2}\).

Step4: Differentiate the area formula with respect to time \(t\)

Using the chain - rule \(\frac{dA}{dt}=\frac{d}{dt}(\pi r^{2})\). By the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\), we have \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\).

Answer:

Circumference: \(C = 2\pi r\)
Change of Circumference: \(\frac{dC}{dt}=2\pi\frac{dr}{dt}\)
Area: \(A=\pi r^{2}\)
Change of Area: \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\)