![MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING<br />Figure depicts the various ways in which an average man gains and loses water in one day. One liter in ingested as food, and the body metabolically produces 0.3L. In breathing air, the exchange is 0.05L while inhaling, and 0.4L while exhaling over a one-day period. The body will also lose 0.2, 1.4, 0.2, and 0.35L through sweat, urine, feces, and through the skin, respectively. In order to maintain steady-state condition, how much water must be drunk per day?<br />FecesAirUrineSkinSweat<br />Food<br />Drink<br />Metabolism<br />For free-falling parachutist with linear drag, assume a first jumper is 70kg and has a drag coefficient of 12kg/s. If a second jumper has a drag coefficient of 15kg/s and a mass of 75kg, how long will it take him to reach the same velocity the first jumper reached in 10s?<br />The amount of a uniformly distributed radioactive contaminant contained in a closed reactor is measured by its concentration c (Becquerel/liter of Bq/L). The contaminant decreases at a decay rate proportional to its concentration-that is <br />Decay rate = -kc<br />where k is a constant with units of day-1. Therefore, a mass balance for the reactor can be written as<br />dcdt = -kc<br />change in mass= decreaseby decay<br />Use Euler’s method to solve this equation from t=0 to 1d with k=0.2d-1. Employ a step size of Δt=0.1. The concentration at t=0 is 10Bq/L.<br />Plot the solution on a semilog graph (i.e., ln c versus t) and determine the slope. Interpret your results.<br />Newton’s law of cooling says that the temperature of a body changes at a rate proportional to the difference between its temperature and that of the surrounding medium (the ambient temperature), <br />dTdt= -k(T-Ta) <br />Where T= the temperature of the body (°C), t= time (min), k= the proportionality constant (per minute), and Ta= the ambient temperature (°C). Suppose that a cup of coffee originally has a temperature of 68°C. Use Euler’s method to compute the temperature from t = 0 to 10 min using a step size of 1min if Ta=21°C and k=0.017/min.<br />Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area<br />dVdt = -kA<br />Where the volume (mm3), t = time (h), k= the evaporation rate (mm/hr), and A= surface area (mm2). Use Euler’s method to compute the volume of the droplet from t = 0 to 10min using step size of 0.25min. Assume that k = 0.1mm/min and that the droplet initially has a radius of 3mm. Assess the validity of your results by determining the radius of your final computed volume and verifying that is consistent with the evaporation rate.<br />A storage tank contains a liquid at depth y where y=0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Qsin2(t).<br />d(Ay)dx=3Qsin3t- Q<br />change involume=inflow- (outflow)<br />Or, since the surface area A is constant<br />dydx=3QAQsin2t- QA<br />Use Euler’s method to solve for the depth y from t = 0 to 10d with a step size of 0.5d. The parameter values are A = 1200 m2 and Q = 500 m3/d. Assume that the initial condition is y = 0.<br />Approximations and Round-off errors<br />Convert the following base-2 numbers to base-10: (a) 101101, (b) 101.101, and (c) 0.01101.<br />Evaluate e-5 using two approaches<br />e-x=1-x+x22-x33!+…<br />and<br />e-x=1e-x=11+x+x22-x33!+…<br />And compare with the true value of 6.737947x10-3. Use 20 terms to evaluate each and compute true and approximate relative errors as terms are added.<br />(a) Evaluate the polynomial<br />Y=x3-7x2+8x-0.35<br />At x=1.37. Use 3-digit arithmetic with chopping. Evaluate the percent relative error.<br />(b) Repeat (a) but express y as<br />y=((x-7)x +8)x - 0.35<br />Evaluate the error and compare with part (a)<br />Determine the number of terms necessary to approximate cos x to 8 significant figures using the Maclaurin series approximation<br />cosx=1-x22+x44!-x66!+x88!-…<br />Calculate the approximation using a value of x = 0.3π. Write a program to determine your result.<br />How can the machine epsilon be employed to formulate a stopping criterion εs for your programs? Provide an example.<br />The infinite series<br />fn=i=1n1/i4<br />Converge on a value of f(n) = π4/90 as n approaches infinity. Write a program in single precision to calculate f(n) for n= 10 000 by computing the sum from i=1 to 10 000. Then repeat the calculation but in reverse order-that is, from i = 10 000 to 1 using increments of -1. In each case, compute the true percent relative error. Explain the results.<br />Truncation Errors and the Taylor Series<br />Use zero- trhough third-order Taylor series expansions to predict f(3) for<br />f(x) = 25x3 – x2 + 7x – 88<br />using a base point at x = 1. Compute the true percent relative error εT for each approximation. Discuss the meaning of the results.<br />Use forward and backward difference approximations of O(h2) to estimate the first derivate of the function examined in before exercise. Perform the evaluation at x=2 using steps sizes of h=0.25 and 0.125. Compare your estimates with the true of the second derivative. <br />Evaluate and interpret the condition numbers for<br />F(x) = |x-1|+1 for x = 1.00001<br />F(x) = e-x for x = 10<br />F(x) = x2+1-xfor x = 200<br />F(x) = e-x-1/x for x = 0.001<br />A missile leaves the ground with an initial velocity Vo forming an angle Φo with the vertical as shown in figure. The maximum desired altitude is αR where R is the radius of the earth. The laws of mechanics can be used to show that<br />sinΦo=1+α1-α1+α vevo2<br />Where ve = the escape velocity of the missile. It is desired to fire the missile and reach the design maximum altitude within an accuracy of +/- 2%. Determine the range of values for Φo if ve/vo = 2 and α=0.25<br />Φo<br />vo<br />R <br />Consider the function f(x) =x3–2x+4 on the interval [-2,2] with h=0.25. Use the forward, backward, and centered finite difference approximations for the first and second derivates so as to graphically illustrate which approximations along with the theoretical, and do the same for the second derivative as well.<br />](https://image.slidesharecdn.com/problems-100726220036-phpapp02/85/Problems-1-320.jpg)
Problems
- 1. MATHEMATICAL MODELING AND ENGINEERING PROBLEM SOLVING<br />Figure depicts the various ways in which an average man gains and loses water in one day. One liter in ingested as food, and the body metabolically produces 0.3L. In breathing air, the exchange is 0.05L while inhaling, and 0.4L while exhaling over a one-day period. The body will also lose 0.2, 1.4, 0.2, and 0.35L through sweat, urine, feces, and through the skin, respectively. In order to maintain steady-state condition, how much water must be drunk per day?<br />FecesAirUrineSkinSweat<br />Food<br />Drink<br />Metabolism<br />For free-falling parachutist with linear drag, assume a first jumper is 70kg and has a drag coefficient of 12kg/s. If a second jumper has a drag coefficient of 15kg/s and a mass of 75kg, how long will it take him to reach the same velocity the first jumper reached in 10s?<br />The amount of a uniformly distributed radioactive contaminant contained in a closed reactor is measured by its concentration c (Becquerel/liter of Bq/L). The contaminant decreases at a decay rate proportional to its concentration-that is <br />Decay rate = -kc<br />where k is a constant with units of day-1. Therefore, a mass balance for the reactor can be written as<br />dcdt = -kc<br />change in mass= decreaseby decay<br />Use Euler’s method to solve this equation from t=0 to 1d with k=0.2d-1. Employ a step size of Δt=0.1. The concentration at t=0 is 10Bq/L.<br />Plot the solution on a semilog graph (i.e., ln c versus t) and determine the slope. Interpret your results.<br />Newton’s law of cooling says that the temperature of a body changes at a rate proportional to the difference between its temperature and that of the surrounding medium (the ambient temperature), <br />dTdt= -k(T-Ta) <br />Where T= the temperature of the body (°C), t= time (min), k= the proportionality constant (per minute), and Ta= the ambient temperature (°C). Suppose that a cup of coffee originally has a temperature of 68°C. Use Euler’s method to compute the temperature from t = 0 to 10 min using a step size of 1min if Ta=21°C and k=0.017/min.<br />Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area<br />dVdt = -kA<br />Where the volume (mm3), t = time (h), k= the evaporation rate (mm/hr), and A= surface area (mm2). Use Euler’s method to compute the volume of the droplet from t = 0 to 10min using step size of 0.25min. Assume that k = 0.1mm/min and that the droplet initially has a radius of 3mm. Assess the validity of your results by determining the radius of your final computed volume and verifying that is consistent with the evaporation rate.<br />A storage tank contains a liquid at depth y where y=0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Qsin2(t).<br />d(Ay)dx=3Qsin3t- Q<br />change involume=inflow- (outflow)<br />Or, since the surface area A is constant<br />dydx=3QAQsin2t- QA<br />Use Euler’s method to solve for the depth y from t = 0 to 10d with a step size of 0.5d. The parameter values are A = 1200 m2 and Q = 500 m3/d. Assume that the initial condition is y = 0.<br />Approximations and Round-off errors<br />Convert the following base-2 numbers to base-10: (a) 101101, (b) 101.101, and (c) 0.01101.<br />Evaluate e-5 using two approaches<br />e-x=1-x+x22-x33!+…<br />and<br />e-x=1e-x=11+x+x22-x33!+…<br />And compare with the true value of 6.737947x10-3. Use 20 terms to evaluate each and compute true and approximate relative errors as terms are added.<br />(a) Evaluate the polynomial<br />Y=x3-7x2+8x-0.35<br />At x=1.37. Use 3-digit arithmetic with chopping. Evaluate the percent relative error.<br />(b) Repeat (a) but express y as<br />y=((x-7)x +8)x - 0.35<br />Evaluate the error and compare with part (a)<br />Determine the number of terms necessary to approximate cos x to 8 significant figures using the Maclaurin series approximation<br />cosx=1-x22+x44!-x66!+x88!-…<br />Calculate the approximation using a value of x = 0.3π. Write a program to determine your result.<br />How can the machine epsilon be employed to formulate a stopping criterion εs for your programs? Provide an example.<br />The infinite series<br />fn=i=1n1/i4<br />Converge on a value of f(n) = π4/90 as n approaches infinity. Write a program in single precision to calculate f(n) for n= 10 000 by computing the sum from i=1 to 10 000. Then repeat the calculation but in reverse order-that is, from i = 10 000 to 1 using increments of -1. In each case, compute the true percent relative error. Explain the results.<br />Truncation Errors and the Taylor Series<br />Use zero- trhough third-order Taylor series expansions to predict f(3) for<br />f(x) = 25x3 – x2 + 7x – 88<br />using a base point at x = 1. Compute the true percent relative error εT for each approximation. Discuss the meaning of the results.<br />Use forward and backward difference approximations of O(h2) to estimate the first derivate of the function examined in before exercise. Perform the evaluation at x=2 using steps sizes of h=0.25 and 0.125. Compare your estimates with the true of the second derivative. <br />Evaluate and interpret the condition numbers for<br />F(x) = |x-1|+1 for x = 1.00001<br />F(x) = e-x for x = 10<br />F(x) = x2+1-xfor x = 200<br />F(x) = e-x-1/x for x = 0.001<br />A missile leaves the ground with an initial velocity Vo forming an angle Φo with the vertical as shown in figure. The maximum desired altitude is αR where R is the radius of the earth. The laws of mechanics can be used to show that<br />sinΦo=1+α1-α1+α vevo2<br />Where ve = the escape velocity of the missile. It is desired to fire the missile and reach the design maximum altitude within an accuracy of +/- 2%. Determine the range of values for Φo if ve/vo = 2 and α=0.25<br />Φo<br />vo<br />R <br />Consider the function f(x) =x3–2x+4 on the interval [-2,2] with h=0.25. Use the forward, backward, and centered finite difference approximations for the first and second derivates so as to graphically illustrate which approximations along with the theoretical, and do the same for the second derivative as well.<br />