Question

a circle is growing, its radius increasing by 3 mm per second. find the rate at which the area is changing at the moment when the radius is 26 mm
when the radius is 26 mm, the area is changing at approximately
(round to the nearest thousandth as needed)

Explanation:

Step1: Write the formula for the area of a circle

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

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

Using the chain - rule \(\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}\).
First, find \(\frac{dA}{dr}\): \(\frac{dA}{dr} = 2\pi r\).
We are given that \(\frac{dr}{dt}=3\) mm/s.

Step3: Substitute the values of \(r\) and \(\frac{dr}{dt}\)

When \(r = 26\) mm and \(\frac{dr}{dt}=3\) mm/s.
\(\frac{dA}{dt}=(2\pi r)\cdot\frac{dr}{dt}\).
Substitute \(r = 26\) and \(\frac{dr}{dt}=3\) into the formula: \(\frac{dA}{dt}=2\pi\times26\times3\).
\(\frac{dA}{dt}=156\pi\).

Step4: Calculate the numerical value

\(\frac{dA}{dt}=156\pi\approx156\times3.14159265 = 490.088\) (rounded to the nearest thousandth)

Answer:

\(490.088\)