The Deepwater Horizon oil spill in the Gulf of Mexico on April 20, 2010 was "an environmental disaster of unprecedented proportions" and was a "devastating blow to the resource-dependent economy of the region" [1]. According to the Washington Post, it is estimated that $9857$ cubic meters of oil per day spilled into the Gulf of Mexico on August 2, 2010 [2].

Assume that an average of $9857 \ \mathrm{m^3/day}$ of oil spilled into the Gulf of Mexico every day and that it formed a hemispherical dome of radius $r$ on the ocean floor.

1. Find a formula for the volume of oil spilled, in cubic meters, $t$ days after the start of the oil spill. Include units in your answer.
   $V(t) =$ help (units)
   
   
2. Find the derivative of your volume formula with respect to time. Include units in your answer.
   $\displaystyle \frac{dV}{dt} =$ help (units)
   
   
3. When the volume of the oil spill is $50,000 \ \mathrm{m^3}$, what is the radius of the hemisphere of oil? Include units in your answer. Give your answer accurate to 3 decimal places.
   $r =$ help (units)
   
   
4. Find the rate of change of the radius with respect to time when the volume of the oil spill is $50,000 \ \mathrm{m^3}$. Include units in your answer.
   $\displaystyle \frac{dr}{dt} =$ help (units)
   
   
5. Since only the oil that is in contact with seawater can mix with seawater, it is important to know how much of the surface area of the oil spill is in contact with seawater. Find the rate of change of the hemisphere's surface area with respect to time when the volume of the oil spill is $50,000 \ \mathrm{m^3}$. Include units in your answer. You should assume that only the top of the hemispherical dome of oil comes in contact with seawater (not the flat bottom, which is incontact with the ocean floor).
   $\displaystyle \frac{dA}{dt} =$ help (units)
   
   
6. Is $\displaystyle A'(t) = \frac{dA}{dt}$ a constant function? If it is not, is it an increasing or decreasing function of time? In other words, as $t$ increases, does $\displaystyle \frac{dA}{dt}$ stay the same, increase, or decrease? Explain and justify your answer using complete sentences with correct grammar, spelling, and punctuation. Please type your answer in the box below.
   
   

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References:

[1] Deepwater Horizon Oil Spill, Phase I Early Restoration Plan and Environmental Assessment, Prepared by the Deepwater Horizon Natural Resource Trustees from: State of Alabama, State of Florida, State of Louisiana, State of Mississippi, State of Texas, Department of the Interior, National Oceanic and Atmospheric Administration. link Retrieved Sept. 15, 2012.

[2] Joel Achenbach and David Fahrenthold (2010-08-02). "Oil well spilled out 4.9 million barrels, new numbers reveal". Washington Post. Retrieved 2010-05-25.