In 1973 an oil barge collided with a bridge in the Mississippi River, leaking oil into the water at a rate estimated at 50 gallons per minute. In 1989 the Exxon Valdez spilled an estimated 11,000,000 gallons of oil into Prudhoe Bay in 6 hours , and in 2010 the Deepwater Horizon well leaked into the Gulf of Mexico at a rate estimated to be 15,000 barrels per day" (1 barrel = 42 gallons). In this project you are going to use differential equations to analyze a simplified model of the dissipation of heavy crude oil spilled at a rate of $ ft'/sec into a flowing body of water. The flow region is a canal, namely a straight channel of rectangular cross section, w feet wide by d feet deep, having a constant flow rate of v fu/sec; the oil is presumed to float in a thin layer of thickness s (feet) on top of the water, without mixing. In Figure 2.12, the oil that passes through the cross-section window in a short time Ar occur pies a box of dimensions s by w by var. To make the analysis easier, presume that the canal is conceptually partitioned into cells of length L ft. each, and that within each particular cell the oil instantaneously disperses and forms a uniform layer of thickness s,(t) in cell i (cell 1 starts at the point of the spill). So, at time t, the ith cell contains s (t)wL ft of oil. Oil flows out of cell i at a rate equal to s,(r) wv ft /sec, and it flows into cell i at the rate s;-() wv; it flows into the first cell at S ft /sec. d Figure 2.12 Oil leak in a canal.(a) Formulate a system of differential equations and initial conditions for the oil thickness in the first three cells. Take S = 50 gallons/min, which was roughly the spillage rate for the Mississippi River incident, and take w = 200 ft, d = 25 ft, and v = 1 mi/hr (which are reasonable estimates for the Mississippi River ). Take _ = 1000 ft. (b) Solve for s, (). [Caution: Make sure your units are consistent.] (c) If the spillage lasts for 7 seconds, what is the maximum oil layer thickness in cell 19 (d) Solve for s,(). What is the maximum oil layer thickness in cell 2? (e) Probably the least tenable simplification in this analysis lies in regarding the layer thickness as uniform over distances of length L. Reevaluate your answer to part (c) with L reduced to 500 ft. By what fraction does the answer change