Question

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=pi r^{2} ). in 2013, the radius of the ice cap was approximately 772 mi and was shrinking at a rate of approximately 4.6 mi/yr. how fast was the area changing at that time? the area was changing at a rate of ( square ) in 2013. (round to the nearest integer as needed.)

Explanation:

Step1: Differentiate the area formula

The area formula is \(A = \pi r^{2}\). Differentiating both sides with respect to time \(t\) using the chain - rule \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\).

Step2: Substitute the given values

We are given that \(r = 772\) mi and \(\frac{dr}{dt}=- 4.6\) mi/yr (negative because the radius is shrinking).
Substitute these values into the formula \(\frac{dA}{dt}=2\pi\times772\times(-4.6)\).
First, calculate \(2\times772\times4.6 = 2\times772\times4.6=772\times9.2 = 7082.4\).
Then \(\frac{dA}{dt}=- 7082.4\pi\).
Using \(\pi\approx3.14\), we have \(\frac{dA}{dt}\approx-7082.4\times3.14=- 22238.736\).

Answer:

\(-22239\)