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| m101220 | The data tabulated below were generated from an experiment initially containing pure ammonium cyanate (NH4OCN). It is known that such concentration changes can be modeled by the following equation:
c = c0 / 1 +kc0t
where c0 and k are parameters. Use a transformation to linearize this equation. Then use linear regression to predict the concentration at t = 160 min. |
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| m101224 | The displacement of a uniform membrane subject to a tension and a uniform pressure can be described by the Poisson equation
Solve for the displacement of a 1-cm-square membrane that has P/T = 0.6/cm and is fastened so that it has zero displacement along its four boundaries. Employ Δx = Dy = 0.1 cm. Display your results as a contour plot. |
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| m101270 | The following data come from a table that was measured with high precision. Use the best numerical method (for this type of problem) to determine y at x = 3.5. Note that a polynomial will yield an exact value. Your solution should prove that your result is exact. |
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| m101271 | The following data define the sea-level concentration of dissolved oxygen for fresh water as a function of temperature:
Estimate o(27) using
(a) Linear interpolation,
(b) Newton s interpolating polynomial, and
(c) Cubic splines. Note that the exact result is 7.986 mg/L? |
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| m101272 | The following data was collected for a cross-section of a river (y = distance from bank, H = depth, and U = velocity):
Use numerical integration to compute the (a) average depth, (b) cross-sectional area, (c) average velocity, and (d) the flow rate. Note that the cross-sectional area (Ac) and the flow rate (Q) can be computed as |
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| m101273 | The following data were collected for the distance traveled versus time for a rocket:
Use numerical differentiation to estimate the rocket s velocity and acceleration at each time. |
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| m101274 | The following differential equation describes the steady state concentration of a substance that reacts with first-order kinetics in an axially-dispersed plug-flow reactor
Where D = the dispersion coefficient (m2/hr), c = concentration (mol/L), x = distance (m), U = the velocity (m/hr), and k = the reaction rate (/hr). The boundary conditions can be formulated as
Ucin = Uc(x = 0) = D(dc/dx) (x = 0)
dc/dx(x = L) = 0
Where cin = the concentration in the inflow (mol/L), and L = the length of the reactor (m). These are called Danckwerts boundary conditions. Use the finite-difference approach to solve for concentration as a function of distance given the following parameters:
D = 5000 m2/hr, U = 100 m/hr, k = 2/hr, L = 100 m, and cin = 100 mol/L. Employ centered finite-difference approximations with ∆x = 10 m to obtain your solutions. Compare your numerical results with the analytical solution, |
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| m101275 | The following information is available for a bank account:
Note that the money earns interest which is computed as
Interest = iBi
where i = the interest rate expressed as a fraction per month, and Bi the initial balance at the beginning of the month.
(a) Use the conservation of cash to compute the balance on 6/1, 7/1, 8/1, and 9/1 if the interest rate is 1% per month (i = 0.01/month). Show each step in the computation.
(b) Write a differential equation for the cash balance in the form
where t = time (months), D(t) = deposits as a function of time ($/month), W(t) = withdrawals as a function of time ($/month).
For this case, assume that interest is compounded continuously; that is, interest = iB.
(c) Use Euler s method with a time step of 0.5 month to simulate the balance. Assume that the deposits and withdrawals are applied uniformly over the month.
(d) Develop a plot of balance versus time for (a) and (c). |
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| m101276 | The following is the built-in humps function that MATLAB uses to demonstrate some of its numerical capabilities:
The humps function exhibits both fl at and steep regions over a relatively short x range. Generate values of this function at intervals of 0.1 over the range from x = 0 to 1. Fit these data with a cubic spline and create a plot comparing the fit with the exact humps function? |
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| m101277 | The following model is frequently used in environmental engineering to parameterize the effect of temperature T (°C) on biochemical reaction rates k (per day),
k = k20θT-20
where k20 and u are parameters. Use a transformation to linearize this equation. Then employ linear regression to estimate k20 and u and predict the reaction rate at T = 17°C. |
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| m101278 | The following ODEs have been proposed as a model of an epidemic:
Where S = the susceptible individuals, I = the infected, R = the recovered, a = the infection rate, and r = the recovery rate. A city has 10,000 people, all of whom are susceptible.
(a) If a single infectious individual enters the city at t = 0, compute the progression of the epidemic until the number of infected individuals falls below 10. Use the following parameters:
a = 0.002/(person?week) and r = 0.15yd. Develop time-series plots of all the state variables. Also generate a phaseplane plot of S versus I versus R.
(b) Suppose that after recovery, there is a loss of immunity that causes recovered individuals to become susceptible. This reinfection mechanism can be computed as rR, where r = the reinfection rate. Modify the model to include this mechanism and repeat the computations in (a) using r = 0.015yd. |
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| m101279 | The following relationships can be used to analyze uniform beams subject to distributed loads,
Where x = distance along beam (m), y = deflection (m), u(x) = slope (m/m), E = modulus of elasticity (Pa = N/m2), I = moment of inertia (m4), M(x) = moment (N m), V(x) 5 shear (N), and w(x) = distributed load (N/m). For the case of a linearly increasing load (recall Fig. P8.18), the slope can be computed analytically as
Employ (a) numerical integration to compute the deflection (in m) and (b) numerical differentiation to compute the moment (in N m) and shear (in N). Base your numerical calculations on values of the slope computed with Eq. P24.19 at equally-spaced intervals of ∆x = 0.125 m along a 3-m beam. Use the following parameter values in your computation: E 5 200 GPa, I = 0.0003 m4, and w0 = 2.5 kN/cm. In addition, the deflections at the ends of the beam are set at y(0) = y(L) = 0. Be careful of units. |
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| m101280 | The general form for a three-dimensional stress field is given by
where the diagonal terms represent tensile or compressive stresses and the off-diagonal terms represent shear stresses. A stress field (in MPa) is given by
To solve for the principal stresses, it is necessary to construct the following matrix (again in MPa):
σ1, σ2, and σ3 can be solved from the equation
σ3 - Iσ2 + IIσ - III = 0
where
I = σxx + σyy + σzz
II = σxxσyy + σxxσzz + σyyσzz - σ2xy - σ2xz - σ2yz
III = σxxσyyσzz - σxxσ2 yz - σyyσ2 xz - σzzσ2xy + 2σxy σxz σyz
I, II, and III are known as the stress invariants. Find σ1, σ2, and σ3 using a root-finding technique. |
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| m101284 | The growth of floating, unicellular algae below a sewage treatment plant discharge can be modeled with the following simultaneous ODEs:
Where t = travel time (d), a = algal chlorophyll concentration (µgA/L), n = inorganic nitrogen concentration (µgN/L), p = inorganic phosphorus concentration (µgP/L), c = detritus concentration (µgC/L), kd = algal death rate (/d), ks = algal settling rate (/d), kh = detrital hydrolysis rate (/d), rnc = nitrogen-to-carbon ratio (µgN/µgC), rpc = phosphorus-to-carbon ratio (µgP/µgC), rna = nitrogen-to-chlorophyll ratio (µgN/µgA), rpa = phosphorus-tochlorophyll ratio (µgP/µgA), and kg(n, p) = algal growth rate (/d), which can be computed with
Where kg = the algal growth rate at excess nutrient levels (/d), ksp = the phosphorus half-saturation constant (µgP/L), and ksn = the nitrogen half-saturation constant (µgN/L). Use the ode45 and ode15s functions to solve these equations from t = 0 to 50 d given the initial conditions a = 1, n = 4000, p = 800, and c = 0. |
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| m101285 | The height of a small rocket y can be calculated as a function of time after blastoff with the following piecewise function:
Y = 38.1454t + 0.13743t3 0 ≤ t < 15
y = 1036 + 130.909(t - 15) 16.18425(t - 15)2
-0.428(t - 15)3 15 ≤ t < 33
y = 2900 - 62.468(t - 33)2 16.9274(t - 33)2
10.4 + 796(t - 33)3 t < 33
Develop a well-structured pseudocode function to compute y as a function of t. Note that if the user enters a negative value of t or if the rocket has hit the ground (y ≤ 0) then return a value of zero for y. Also, the function should be invoked in the calling program as height(t). Write the algorithm as (a) pseudocode, or (b) in the high-level language of your choice. |
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| m101290 | The length of the longest ladder that can negotiate the corner depicted in Fig. P15.17 can be determined by computing the value of u that minimizes the following function:
For the case where w1 = w2 = 2 m, use a numerical method (including software) to develop a plot of L versus a range of α s from 45o to 135o.
Figure P15.17
A ladder negotiating a corner formed by two hallways. |
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| m101291 | The Lotka-Volterra equations described in Sec. 28.2 have been refined to include additional factors that impact predator-prey dynamics. For example, over and above predation, prey population can be limited by other factors such as space. Space limitation can be incorporated into the model as a carrying capacity (recall the logistic model described in Prob. 28.16) as in
Where K = the carrying capacity. Use the same parameter values and initial conditions as in Sec. 28.2 to integrate these equations from t 5 0 to 100 using ode45.
(a) Employ a very large value of K = 108 to validate that you obtain the same results as in Sec. 28.2.
(b) Compare (a) with the more realistic carrying capacity of K = 200. Discuss your results. |
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| m101293 | The Manning equation can be written for a rectangular open channel as
where Q = flow [m3/s], S 5 slope [m/m], H = depth [m], and n = the Manning roughness coefficient. Develop a fixed-point iteration scheme to solve this equation for H given Q = 5, S = 0.0002, B = 20, and n = 0.03. Prove that your scheme converges for all initial guesses greater than or equal to zero. |
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| m101312 | The Poisson equation can be written in three dimensions as
Solve for the distribution of temperature within a unit cube with zero boundary conditions and f = -10. Employ ∆x = ∆y = ∆z 1/6. |
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| m101316 | The pseudocode in Fig. P2.25 computes the factorial. Express this algorithm as a well-structured function in the language of your choice. Test it by computing 0! and 5!. In addition, test the error trap by trying to evaluate 22!. |
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