In a trend that scientists​ attribute, at least in​ part, to global​ warming, a certain floating cap of sea ice has been shrinking since 1980. The ice cap always shrinks in the summer and grows in winter. Average minimum size of the ice​ cap, in square​ miles, can be approximated by A=πr^2. In 2013,the radius of the ice cap was approximately 785 mi and was shrinking at a rate of approximately 4.3 mi/yr. How fast was the area changing at that​ time?

To solve this problem, we can use related rates, since we're given how fast the radius of the ice cap is changing and need to determine how fast the area is changing.

### Step 1: Formula for the area of the ice cap

The area \( A \) of the ice cap is approximated by the formula for the area of a circle:

\[
A = \pi r^2
\]

where \( A \) is the area of the ice cap and \( r \) is the radius of the ice cap.

### Step 2: Differentiate the area with respect to time

We want to find how fast the area is changing, i.e., \( \frac{dA}{dt} \). To do this, we take the derivative of the area formula with respect to time \( t \):

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

Using the chain rule:

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

where \( \frac{dr}{dt} \) is the rate at which the radius is shrinking.

### Step 3: Plug in the given values

We are given:
- The radius \( r = 785 \) miles in 2013.
- The rate of change of the radius \( \frac{dr}{dt} = -4.3 \) mi/yr (negative because the radius is shrinking).

Now, substitute these values into the equation:

\[
\frac{dA}{dt} = 2\pi (785)(-4.3)
\]

### Step 4: Calculate the rate of change of the area

Let's compute \( \frac{dA}{dt} \).

\[
\frac{dA}{dt} = 2\pi (785)(-4.3) = -2 \times 3.1416 \times 785 \times 4.3
\]

I'll calculate this next.The area of the ice cap was shrinking at a rate of approximately \(-21,209\) square miles per year in 2013.

Would you like further details or have any questions? Here are five related questions to expand your understanding:

1. How would the rate of area change be affected if the shrinking rate of the radius were halved?
2. What would happen to the area if the radius remained constant over time?
3. How would you solve this if the shrinking rate of the radius were not constant?
4. Can you relate this problem to a real-life environmental impact of shrinking sea ice?
5. What would be the radius of the ice cap after 10 years, given the same rate of shrinkage?

**Tip:** In related rates problems, remember to differentiate both sides of the equation with respect to time and use the chain rule when necessary.

In a trend that scientists attribute, at least in part, to global warming, a certain floating cap of sea ice has been shrinking since 1980. The ice cap always shrinks in the summer and grows in winter. Average minimum size of the ice cap, in square miles, can be approximated by A=πr^2. In 2013, the radius of the ice cap was approximately 785 mi and was shrinking at a rate of approximately 4.3 mi/yr. How fast was the area changing at that time?

This problem involves calculating the rate at which the area of a shrinking ice cap is changing using related rates in calculus. With a radius of 785 miles and shrinking at 4.3 mi/yr, find the rate of change of the area using the formula A = πr^2. Suitable for college-level calculus students.

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