Questions: 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 21 mm. When the radius is 21 mm, the area is changing at approximately (Round to the nearest thousandth as needed.)

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 21 mm.

When the radius is 21 mm, the area is changing at approximately
(Round to the nearest thousandth as needed.)

Transcript text: 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 21 mm . When the radius is 21 mm , the area is changing at approximately $\square$ $\square$ (Round to the nearest thousandth as needed.)

Solution

Solution Steps

Step 1: Express the area of the circle as a function of its radius

The area $A$ of a circle is given by the formula $A = \pi r^2$.

Step 2: Apply the chain rule of differentiation

To find the rate at which the area is changing, we use the chain rule: $ rac{dA}{dt} = rac{dA}{dr} \cdot rac{dr}{dt}$.

Step 3: Differentiate the area function with respect to $r$

Differentiating $A = \pi r^2$ with respect to $r$ gives $ rac{dA}{dr} = 2\pi r$.

Step 4: Substitute \(

rac{dA}{dr}\) and $ rac{dr}{dt}$ into the formula Substituting $ rac{dA}{dr} = 2\pi r$ and $ rac{dr}{dt} = 3$ into the formula gives $ rac{dA}{dt} = 2\pi \cdot 21 \cdot 3 = 395.84$.