in a trend that scientists attribute, at least in part, to global warming, a certain floating cap of sea ice…

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 779 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)

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 779 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)

Answer

Explanation:

Step1: Differentiate the area formula with respect to time

Given (A = \pi r^{2}), using the chain - rule (\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}). Differentiate (A) with respect to (r): (\frac{dA}{dr} = 2\pi r). So (\frac{dA}{dt}=2\pi r\frac{dr}{dt}).

Step2: Substitute the given values

We know that (r = 779) mi and (\frac{dr}{dt}=- 4.8) mi/yr (negative because the radius is shrinking). Substitute these values into the formula (\frac{dA}{dt}): (\frac{dA}{dt}=2\pi\times779\times(-4.8)). First, calculate (2\times779\times4.8 = 2\times3739.2=7478.4). Then (\frac{dA}{dt}=-\pi\times7478.4).

Step3: Calculate the numerical value

(\frac{dA}{dt}\approx - 3.14\times7478.4=-23482.176)

Answer:

(-23482)