3. Pulling Gasoline Up (Work on Fluids) A cylindrical tank with a length of 10 m and a radius of 5 m is on its side and half-full of diesel (see figure below). A vacuum pump is lifting the "handfuls" of diesel to the top of the tank, so that the diesel level changes from 0 to -5. length=10 D(y) y- width width of the gasoline layer y-0 area of the gasoline layer How much work is required to empty the tank through an outlet pipe at the top of the tank? The equation of work for this type of problems is: W = LpgA(y)D(y)dy Where: p = 737 kg/m³ is the density of gasoline. g=9.8 m/s² is the acceleration of gravity. A(y) is the cross-sectional area of the gasoline layer. D(y) is the distance that the slices of gasoline must be lifted. Useful integrals [hint: I recommend you to use MATLAB to avoid mistakes]: √ √ a² = x² dx = sin−1 +√a²x²+c Sx√(a²x²)dx = −(4) Where u = a²x², and thus the new limits become a' = a² - a² and ẞ' = a² - ẞ²

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24. Oil Flow Oil flows through a cylindrical pipe of radius 3 inches, but friction from the pipe slows the flow toward the outer edge. The speed at which the oil flows at a distance r inches from the center is 8(10 - r²) inches per second. (a) In a plane cross section of the pipe, a thin ring with thick- ness Ar at a distance r inches from the center approximates a rectangular strip when you straighten it out. What is the area of the strip (and hence the approximate area of the ring)? (b) Explain why we know that oil passes through this ring at approximately 8(10-2) (2πr) Ar cubic inches per second. (c) Set up and evaluate a definite integral that will give the rate (in cubic inches per second) at which oil is flowing through the pipe.