Problem 5. 17: Ignore the part about performing 3 iterations; rather, run the False-Position algorithm (using a Matlab function) until the approximate relative error is below 0.1%. Print the code and indicate the answer. Then, plot the volume of fluid versus height of fluid (assuming R = 3m), remembering to label axes. Use this plot to check if your answer seems reasonable. Print the plot. where g = 9.81 m/s², H = initial head (m), L = pipe length (m), and elapsed time (s). Determine the head needed to achieve v = 5 m/s in 2.5 s for a 4-m-long pipe (a) graphically, (b) by bisection, and (c) with false position. Employ initial guesses of x = 0 and x,, = 2 m with a stopping criterion of ɛ, = 1%. Check you results. 5.16 Water is flowing in a trapezoidal channel at a rate of Q = 20 m³/s. The critical depth y for such a channel must satisfy the equation 0=1 B 8A where g = 9.81 m/s², A = the cross-sectional area (m²), and B = the width of the channel at the surface (m). For this case, the width and the cross-sectional area can be related to depth y by B=3+ y and A = 3y+ 2 Solve for the critical depth using (a) the graphical method, (b) bisec- tion, and (c) false position. For (b) and (c) use initial guesses of x₁ = 0.5 and x = 2.5, and iterate until the approximate error falls below 1% or the number of iterations exceeds 10. Discuss your results. 5.17 You are designing a spherical tank (Fig. P5.17) to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as V = Th² 2 [3R – h] 3 where V = volume (m³), h = depth of water in tank (m), and R = the tank radius (m). your answer. Determine the approximate relative error after each iteration. Employ initial guesses of 0 and R. 5.18 The saturation concentration of dissolved oxygen in freshwa- ter can be calculated with the equation (APHA, 1992) Ino -139.34411 + 1.575701 x 105 Ta 6.642308 x 107 T 1.243800X1010 T 8.621949 X 10" T where of the saturation concentration of dissolved oxygen in freshwater at 1 atm (mg/L) and T₁ = absolute temperature (K). Remember that T₁ =T+273.15, where T = temperature (°C). According to this equation, saturation decreases with increasing temperature. For typical natural waters in temperate climates, the equation can be used to determine that oxygen concentration ranges from 14.621 mg/L at 0°C to 6.413 mg/L at 40°C. Given a value of oxygen concentration, this formula and the bisection method can be used to solve for temperature in °C. (a) If the initial guesses are set as 0 and 40°C, how many bisection iterations would be required to determine temperature to an absolute error of 0.05°C? (b) Develop and test a bisection program to determine T as a func- tion of a given oxygen concentration to a prespecified absolute error as in (a). Given initial guesses of 0 and 40°C, test your program for an absolute error = 0.05°C and the following cases: o 8, 10, and 12 mg/L. Check your results. 5.19 According to Archimedes principle, the buoyancy force is equal to the weight of fluid displaced by the submerged portion of an object. For the sphere depicted in Fig. P5.19, use bisection to deter- mine the heighth of the portion that is above water. Employ the follow- ing values for your computation: r = 1 m, p, = density of sphere = 200 kg/m³, and p₁ = density of water = 1000 kg/m³. Note that the volume of the above-water portion of the sphere can be computed with FIGURE P5.17 R If R = 3 m, to what depth must the tank be filled so that it holds 30 m³? Use three iterations of the false-position method to determine v= Th² 3 (3r-h) FIGURE P5.19