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 Upper A equals pi font size decreased by 5 r squared. In​ 2013, the radius of the ice cap was approximately 760 mi and was shrinking at a rate of approximately 4.6 mi divided by yr. How fast was the area changing at that​ time?
Question content area bottom
Part 1
The area was changing at a rate of

enter your response here
▼
mi divided by yr
mi squared divided by yr
yr divided by mi squared
mi squared
yr
yr divided by mi
mi
in 2013.
​(Round to the nearest integer as​ needed.)

To find how fast the area of the ice cap is changing, we need to calculate the derivative of the area formula with respect to time.

Given that:
- The area of the ice cap \( A \) is approximated by \( A = \pi r^2 \), where \( r \) is the radius.
- The radius of the ice cap is shrinking, so \( r = 760 \) miles in 2013, and the rate of change of the radius \( \frac{dr}{dt} \) is \( -4.6 \) mi/yr (negative because the radius is shrinking).

### Step 1: Differentiate the Area Formula
To find how fast the area is changing, we take the derivative of the area \( A = \pi r^2 \) with respect to time \( t \):

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

### Step 2: Substitute the Known Values
Now, substitute \( r = 760 \) mi and \( \frac{dr}{dt} = -4.6 \) mi/yr into the equation:

\[
\frac{dA}{dt} = 2\pi (760) (-4.6)
\]

### Step 3: Perform the Calculation

Let's compute this.The area of the ice cap was decreasing at a rate of approximately **21,966 square miles per year** in 2013.

Would you like any more details, or do you have any questions?

Here are five related questions you might explore:
1. How would the rate of area change differ if the radius were shrinking faster or slower?
2. How does the rate of change in area depend on the initial size of the ice cap?
3. What happens to the area if the radius remains constant for a few years?
4. Can this model be applied to predict future ice cap areas if the shrinking trend continues?
5. How would the shrinking of the ice cap impact sea levels?

**Tip:** When solving problems involving related rates, always keep track of units to ensure consistent and correct interpretations.

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 = πr². In 2013, the radius of the ice cap was approximately 760 mi and was shrinking at a rate of approximately 4.6 mi/yr. How fast was the area changing at that time?

Discover how to calculate the rate at which the area of a shrinking ice cap changes over time using related rates and calculus. This problem involves differentiating the area of a circle and applying the chain rule. In 2013, the radius of the ice cap was shrinking at a rate of 4.6 mi/yr, and the ice cap's area was decreasing at approximately 21,966 square miles per year. Suitable for students in Grades 11-12 or AP Calculus.

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