A nutjob tries to profit off of the (briefly) negative crude oil prices from the week of April 20, 2020 by “buying” oil and then dumping it into the Gulf of Mexico. Their tanker has a simple valve at the bottom, so the rate at which oil flows out of their tank depends on time, with the flow being faster at first, then slowing as the tank empties. The oil flows at a rate = 40 -0.021, V(0) = 0, (1) Where V (t) represents the total volume in liters that has flowed out in minutes t since the valve was first opened. It’s expected that once the flow rate goes from being positive to negative, this

A nutjob tries to profit off of the (briefly) negative crude oil prices from the week of April 20, 2020 by “buying” oil and then dumping it into the Gulf of Mexico. Their tanker has a simple valve at the bottom, so the rate at which oil flows out of their tank depends on time, with the flow being faster at first, then slowing as the tank empties. The oil flows at a rate = 40 -0.021, V(0) = 0, (1) Where V (t) represents the total volume in liters that has flowed out in minutes t since the valve was first opened. It’s expected that once the flow rate goes from being positive to negative, this model is no longer valid. (a) What are the units of “? (b) What is at t=0? What does this represent, physically? (Hint: this is not the same thing as V(O).) (c) The description referred to the idea that when V'(t) goes from positive to negative, the model is invalid. Give a short explanation why what makes negative derivatives not make sense in this problem?) After giving reasoning, solve for the time when = 0. (d) Solve the pure time differential equation for VO). By the time the flow has reached zero (what you found in the previous step), what is the total volume in liters that has flowed out of the tank? Hint: remind yourself what V() represents. (e) Use the Fundamental Theorem of Calculus to measure how much oil has spilled in the in the first two hours. Do the same to measure how much oil has spilled from hours two to four. You can leave the final expression unsimplified, but feel free to use a calculator if you want to sanity check your result. (Hint 1: be careful with units and your limits of integration.) (Hint 2: you might find Example 4.5.5 in your book useful if you’d like a similar problem to compare to.)

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