Connected Rates of Change (Fixed)

1. The Concept: The Chain Rule in Time

In "Connected Rates of Change" problems, two variables (like Area and Radius) are linked by a formula, but both are changing over time.

$$ \frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt} $$

1. The Rate you Want
$\frac{dA}{dt}$
(How fast Area changes)

2. The Connection
$\frac{dA}{dr}$
(Differentiate the formula $A=\pi r^2$)

3. The Rate you Know
$\frac{dr}{dt}$
(Given in question)

Key Strategy: Always write down what you Know, what you Want, and the Formula that connects the variables.

Problem: A circle's radius increases at 0.1 cm/s. How fast is the Area growing?

Time ($t$): 0s

Radius ($r$): 2.00 cm

Area ($A$): 12.57 cm²

Calculated Rate $\frac{dA}{dt}$:

$$ 2\pi r \times 0.1 $$

= 1.26 cm²/s

Notice: As the circle gets bigger ($r$ increases), the Area grows faster even though $dr/dt$ is constant.

Let's solve the specific problem from the image.

Question: Find the rate of increase of $A$ when $r = 4$, given $\frac{dr}{dt} = 0.1$.

Step 1: Differentiate the Area Formula $A = \pi r^2$ with respect to $r$.
$\frac{dA}{dr} =$

Step 2: Apply Chain Rule Formula.
$\frac{dA}{dt} = (\frac{dA}{dr}) \times (\frac{dr}{dt})$
$\frac{dA}{dt} = (\dots) \times 0.1$

Step 3: Substitute $r = 4$.
$\frac{dA}{dt} = 2\pi(4) \times 0.1$
$\frac{dA}{dt} =$ $\pi$

Test your understanding. Calculate the rate of change of Area.

Your Answer (in terms of $\pi$):

$\pi$ cm²/s