We are given that the radius $r$ of a circle is decreasing at a constant rate of 7 cm/min, and at the instant when the radius is 33 cm, we need to determine the rate of change of the area.

Step 1: Formula for the area of a circle

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

Step 2: Differentiate the area formula with respect to time $t$

To find how the area is changing with respect to time, we differentiate $A$ with respect to $t$ using the chain rule: $\frac{dA}{dt} = \frac{d}{dt} (\pi r^2)$ By the chain rule: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$

Step 3: Substitute the given values

We are told that:

• The radius is $r = 33$ cm,
• The radius is decreasing, so $\frac{dr}{dt} = -7$ cm/min (the negative sign indicates a decrease).

Substituting these values into the formula for the rate of change of area: $\frac{dA}{dt} = 2\pi (33) (-7)$

Step 4: Calculate the rate of change of the area

$\frac{dA}{dt} = -462\pi \approx -462 \times 3.1416 \approx -1457.106 \text{ cm}^2/\text{min}$

Final Answer:

The rate of change of the area is approximately: $\frac{dA}{dt} \approx -1457.106 \text{ cm}^2/\text{min}$

Let me know if you need further clarification or have more questions! Here are some related questions to deepen your understanding:

1. How does the rate of change of the area change if the radius is increasing instead of decreasing?
2. How would the rate of change of the area be affected if the radius were 50 cm instead of 33 cm?
3. What would happen to the rate of change of the area if the rate of change of the radius were increased to -10 cm/min?
4. How do the concepts of related rates apply in other real-world scenarios, such as the growth of a balloon?
5. What is the relationship between the rate of change of the area and the rate of change of the circumference of the circle?

Tip: Remember, when dealing with related rates problems, it’s crucial to carefully set up your equations first and then differentiate using the chain rule. Always be mindful of the signs, especially when something is increasing or decreasing!