MATLAB QUESTION

Find the solution of the given applications of differential equation using MATLAB.

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Project 2 Designing a Drip Dispenser for a Hydrology Experiment In order to make laboratory measurements of water filtration and saturation rates in various types of soils under the condition of steady rainfall, a hydrologist wishes to design drip dispensing containers in such a way that the water drips out at a nearly constant rate. The containers are supported above glass cylinders that contain the soil samples (Figure 2.P.1). The hydrologist elects to use the following differential equation based on Torricelli's prin ciple to help solve the design problem, 4) --/2g. In Eq. (1), ) is the height of the liquid surface above the dispenser outlet at time, 4) is the cross-sectional area of the dispenser at height I, a is the area of the outlet, and is a measured contraction coefficient that accounts for the observed fact that the Cross section of the (smooth) outfiw stream is smaller than a. Note that the hydrologist is using a laminar flow model as a guide in designing the shape of the container. Forces due to surface tension at the tiny outlet are ignored in the design problem. Once the shape FIGURE 2.P1 Water dripping into a soil sample. of the container has been determined the outlet aperture is adjusted to a desired drip rate that will remain nearly constant for an extended period of time. Of course, since surface tension forces are not accounted for in Eq. (1), the equation is not a valid model for the output flow rate when the aperture is so small that the water drops out. Nevertheless, once the hydrologist sees and interprets the results based on her design strategy, she feels justified in using Eq. (1) the containers is determined by requiring that the initial volume of water satisfies V(0) = = 1 Ar(h) dh = 1 ft. Project 2 PROBLEMS 1. Assume that the shape of the dispensers are surfaces of revolution so that A(h) = A[r(h)]}, where r(h) is the radius of the container at height h. For each of the h-dependent cross-sectional radii prescribed below in (1)-(v), (a) Create a surface plot of the surface of revolu- tion, and (b) Find numerical approximations of solutions of Eq. (1) for 0