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 769 mi and was shrinking at a rate of approximately 4.8 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: Find the radius in 2013

Given \(A = \pi r^{2}\), and \(A = 769\) in 2013. Then \(r^{2}=\frac{A}{\pi}\), so \(r=\sqrt{\frac{769}{\pi}}\approx\sqrt{\frac{769}{3.14}}\approx\sqrt{245}\approx15.65\)

Step2: Differentiate the area formula with respect to time

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

Step3: Substitute the values of \(r\) and \(\frac{dr}{dt}\)

We are given \(\frac{dr}{dt}=- 4.8\) (negative because the radius is shrinking). Substitute \(r\approx15.65\) and \(\frac{dr}{dt}=-4.8\) into \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\)

\(\frac{dA}{dt}=2\times3.14\times15.65\times(-4.8)\)

\(\frac{dA}{dt}=6.28\times15.65\times(-4.8)\)

\(6.28\times15.65 = 98.382\)

\(\frac{dA}{dt}=98.382\times(-4.8)\approx - 472.23\)

Answer:

\(-472\)