(a)
If A is the area of a circle with radius r and the circle expands as time passes, find 
dA
dt
 in terms of 
dr
dt
.
dA
dt
 = 

 
  

 
dr
dt
(b)
Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 2 m/s, exactly how fast (in m2/s) is the area of the spill increasing when the radius is 
34 m?
  m2/s

### (a) Finding \( \frac{dA}{dt} \) in terms of \( \frac{dr}{dt} \):

The area \( A \) of a circle is given by the formula:

\[
A = \pi r^2
\]

To find \( \frac{dA}{dt} \), we differentiate both sides of the equation with respect to time \( t \), applying the chain rule:

\[
\frac{dA}{dt} = \frac{d}{dt} (\pi r^2) = 2\pi r \frac{dr}{dt}
\]

Thus, the rate of change of the area with respect to time is:

\[
\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
\]

### (b) Finding the rate of change of the area when the radius is 34 m:

We are given that the radius \( r \) increases at a constant rate of \( \frac{dr}{dt} = 2 \, \text{m/s} \). We want to find how fast the area of the oil spill is increasing when the radius is \( r = 34 \, \text{m} \).

Using the formula from part (a):

\[
\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
\]

Substitute \( r = 34 \, \text{m} \) and \( \frac{dr}{dt} = 2 \, \text{m/s} \):

\[
\frac{dA}{dt} = 2\pi (34) (2) = 136\pi \, \text{m}^2/\text{s}
\]

Approximating \( \pi \approx 3.1416 \):

\[
\frac{dA}{dt} \approx 136 \times 3.1416 = 427.26 \, \text{m}^2/\text{s}
\]

Thus, the area of the oil spill is increasing at a rate of approximately \( 427.26 \, \text{m}^2/\text{s} \) when the radius is 34 meters.

---

### Do you have any questions or want more details?

Here are 5 related questions for you to explore further:
1. How does the rate of area change depend on the radius if \( \frac{dr}{dt} \) is constant?
2. What happens to \( \frac{dA}{dt} \) if \( \frac{dr}{dt} \) is not constant?
3. Can you find \( \frac{dA}{dt} \) for different shapes, such as a square or rectangle?
4. How does \( \frac{dA}{dt} \) change if \( r \) is expressed as a function of time, \( r(t) \)?
5. How would the solution change if the oil spill expanded in an elliptical pattern?

#### Tip:
Always apply the chain rule when differentiating a function of another variable that depends on time, such as area with respect to radius expanding over time!

If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt. Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 2 m/s, exactly how fast (in m2/s) is the area of the spill increasing when the radius is 34 m?

Understand how to find the rate of change of a circle's area as its radius expands. This problem explains how to apply the chain rule in differentiation to solve for dA/dt when dr/dt is known. Explore a real-world example of an oil spill, where the radius increases at 2 m/s, and find how fast the area is growing when the radius is 34 m. Ideal for high school students studying calculus.

Math Problem Statement

(a) If A is the area of a circle with radius r and the circle expands as time passes, find dA dt in terms of dr dt . dA dt

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