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Design a single-support water tower of maximum height, $h$, and maximum water storage capacity. For this exercise, the only mode of failure to protect against is support column buckling. The basic tower design is shown in Fig. 6.11. (1) The steel water tank is a spherical pressure vessel of thickness $t$. (2) The support column has a circular cross section and is made of steel. (3) The weight of the full tank acts vertically at Point B. Design Variables: (1) Column height: $h$ (2) Tank radius: $r$ Side constraints: (1) Height: $10 \leq h \leq 16$ (meters) (2) Radius: $2.13 \leq r \leq 14$ (meters) Numerical Constants: (1) Modulus of steel, $E=206 \mathrm{GPa}$ (2) Diameter of support column, $d=0.3 \mathrm{~m}$ (3) Thickness of tank, $t=0.0127 \mathrm{~m}$ (4) Buckling safety factor, $\mathrm{SF}=2$ (5) Gravitational constant, $g=9.8 \mathrm{~m} / \mathrm{s}^2$ (6) Density of steel, $\rho_{\mathrm{s}}=7,800 \mathrm{~kg} / \mathrm{m}^3$ (7) Density of water, $\rho_w=1,000 \mathrm{~kg} / \mathrm{m}^3$ The questions are: (a) Find the tallest water tower design, $h$. (b) Find the largest storage capacity design. (c) Find four other Pareto solutions that are significantly different from any other design you have obtained. How do you compare your designs to decide the extent to which they are different? (d) Plot all your designs from the previous parts in the $\mu_1-\mu_2$ space. Turn in your plot and your M-files.

Optimization in Practice with MATLAB®: For Engineering Students and Professionals

The Environmental Protection Agency is investigating an abandoned chemical plant. A large, closed cylindrical tank contains an unknown liquid. You must determine the liquid's density and the height of the liquid in the tank (the vertical distance from the surface of the liquid to the bottom of the tank). To maintain various values of the gauge pressure in the air that is above the liquid in the tank, you can use compressed air. You make a small hole at the bottom of the side of the tank, which is on a concrete platform - so the hole is $50.0 \mathrm{~cm}$ above the ground. The table gives your measurements of the horizontal distance $R$ that the initially horizontal stream of liquid pouring out of the tank travels before it strikes the ground and the gauge pressure $p_{\mathrm{g}}$ of the air in the tank. $$\begin{array}{l|lllll}p_{\mathrm{g}}(\mathrm{atm}) & 0.50 & 1.00 & 2.00 & 3.00 & 4.00 \\\hline \boldsymbol{R}(\mathrm{m}) & 5.4 & 6.5 & 8.2 & 9.7 & 10.9\end{array}$$ (a) Graph $R^{2}$ as a function of $p_{\mathrm{g}}$. Explain why the data points fall close to a straight line. Find the slope and intercept of that line. (b) Use the slope and intercept found in part (a) to calculate the height $h$ (in meters) of the liquid in the tank and the density of the liquid (in $\mathrm{kg} / \mathrm{m}^{3}$ ). Use $g=9.80 \mathrm{~m} / \mathrm{s}^{2} .$ Assume that the liquid is nonviscous and that the hole is small enough compared to the tank's diameter so that the change in $h$ during the measurements is very small.

The Environmental Protection Agency is investigating an abandoned chemical plant. A large, closed cylindrical tank contains an unknown liquid. You must determine the liquid's density and the height of the liquid in the tank (the vertical distance from the surface of the liquid to the bottom of the tank). To maintain various values of the gauge pressure in the air that is above the liquid in the tank, you can use compressed air. You make a small hole at the bottom of the side of the tank, which is on a concrete platform - so the hole is $50.0 \mathrm{~cm}$ above the ground. The table gives your measurements of the horizontal distance $R$ that the initially horizontal stream of liquid pouring out of the tank travels before it strikes the ground and the gauge pressure $p_{\mathrm{g}}$ of the air in the tank. $$\begin{array}{l|lllll}p_{\mathrm{g}}(\mathrm{atm}) & 0.50 & 1.00 & 2.00 & 3.00 & 4.00 \\\hline \boldsymbol{R}(\mathrm{m}) & 5.4 & 6.5 & 8.2 & 9.7 & 10.9\end{array}$$ (a) Graph $R^{2}$ as a function of $p_{\mathrm{g}}$. Explain why the data points fall close to a straight line. Find the slope and intercept of that line. (b) Use the slope and intercept found in part (a) to calculate the height $h$ (in meters) of the liquid in the tank and the density of the liquid (in $\mathrm{kg} / \mathrm{m}^{3}$ ). Use $g=9.80 \mathrm{~m} / \mathrm{s}^{2} .$ Assume that the liquid is nonviscous and that the hole is small enough compared to the tank's diameter so that the change in $h$ during the measurements is very small.

University Physics with Modern Physics

Consider the following bi-objective optimization problem. This is a standard single-variable bi-objective problem often used to test multiobjective optimizers. $$ \begin{gathered} \mu_1=x^2 \\ \mu_2=(x-2)^2 \\ -5 \leq x \leq 5 \end{gathered} $$ (a) Obtain several optimal points on the Pareto frontier using the weighted sum method. Use the MatLaB function fmincon for optimization. Plot each design objective as a function of $x$ on the same figure (as shown in Fig. 6.2). Identify on this plot, the Pareto solutions that you just obtained. Turn in your M-files. (b) Plot the Pareto optimal points in the $\mu_1-\mu_2$ space. Turn in your M-files and the plot.

Optimization in Practice with MATLAB®: For Engineering Students and Professionals

Questions asked

INSTANT ANSWER

Assume residents of Los Angeles receive utility from individual disaster preparedness, 𝑑, and other goods, 𝑥. The price of one unit of disaster preparedness is 𝑝 and the price of other goods is normalized to 1. Also assume that residents receive exogenous income 𝑀. h) Show graphically the resident’s budget constraint under the new transfer program, under the old subsidy program, and under no program at all. Assume each resident’s transfer equals the amount the city gave the resident under the old subsidy program. Again, label your graph, and indicate the optimal choice under the new transfer program as 𝑑𝑇. Does the resident purchase more or fewer units of 𝑑 compared to the subsidy program? Discuss the substitution and income effects created by switching to the transfer program from the subsidy program.

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ANSWERED

Luke Humphrey verified

Numerade educator

Problem 5 (4 points) How would a decision-maker with a single-attribute utility function u2(x) = { (1 - e^-2x) if 0 ? x ? 0.281 e^-3x if x > 0.281 respond to each lottery question? Justify your answer. (i) ?0.1, 0.5, 0? ??? 0.1 (ii) ?0.2, ???, 0.1? ? 0.20 (iii) ????, 0.5, 0.7? ? 0.5 (iv) ?0.6, 0.5, 0.1? ? ???

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ANSWERED

Luke Humphrey verified

Numerade educator

Problem 4 (5 points) A decision maker's utility function for the outcome x is as follows: u(x)=left{egin{array}{ll} (1-e^{-0.2 x}) & ext { if } 0 leq x leq 10 \ 0 & ext { if } x<0, x>10 end{array} ight. The decision maker would like to make a choice between two alternatives A1 and A2. The alternatives result in an uncertain continuous outcome X with the following probability density functions: Alternative A1 : Normal distribution with a mean ?=4.0 and standard deviation of ?=1.0. f(x ; ?, ?)=frac{1}{u0303 sqrt{2 pi}} e^{-frac{1}{2}left(frac{x-u0300}{u0303} ight)^{2}} Alternative A2 : Exponential distribution with

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ANSWERED

Breanna Ollech verified

Numerade educator

Problem 3 (4 points) A decision maker would like to make a choice between two alternatives ( A_{1} ) and ( A_{2} ). The alternatives result in an uncertain discrete outcome ( X ) with binomial probability distribution [ P(x=k)=left(egin{array}{l} n \ k end{array} ight) p^{k}(1-p)^{n-k} ] with the following parameters: - Alternative ( A_{1}: p=0.6, n=40 ) - Alternative ( A_{2}: p=0.4, n=40 ) The decision maker's utilities for the outcomes are given by the following utility function: ( u(x)=left(1-e^{-0.1 x} ight) ) for ( x in{0,1,2, ldots, 40} ). (i) Determine whether one of the alternatives probabilistically dominates the other alternative. (ii) Calculate the expected utility for the two alternatives. Which alternative should the decision maker choose based on the available information?

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ANSWERED

Luke Humphrey verified

Numerade educator

Three suppliers (A, B, and C) produce an automotive part and supply to an automaker. Supplier A produces 30% of the parts. Suppliers B and C produce 50% and 20% of the parts, respectively. 2% of the parts produced by Supplier A, 1% of the produced by Supplier B, and 3% produced by Supplier C are defective. A part is selected at random in the market and found to be defective. What is the probability that this part was produced by: (i) Supplier A? (ii) Supplier B? (iii) Supplier C?

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ANSWERED

Lucas Finney verified

Numerade educator

Problem 1 (3 points) Given P(A) = 0.10, P(B|A) = 0.39, and P (B|A') = 0.39, calculate the following: P(A'), P(B'|A), P(B'|A'), P(B), P(B'), P(A|B), P(A'|B), P(A|B'), and P(A'|B').

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INSTANT ANSWER

Crahlem 5 Applicotion to a Design Probiem (5 points) Design a single-suppart water tower of maximum height, \( h \), and maximum water starage capacity. The basic tower design is shown in the figure below. Assumptions: - The steel water tank is a spherical pressure vessel of thickness \( t \). - The support column has a circular cross-section and is made of steel. - The weight of the full tank acts vertically at Point B. Design Voriables: - Column height: \( h \) - Tank radius: r Variable Bounds: - Height: \( 10 \leq h \leq 16 \) (meters) - Radius: \( 2.13 \leq r \leq 14 \) (meters) Numerical Canstants. - Modulus of elasticity of steel, E = 20652 - Diameter of the support column, \( d=0.3 \mathrm{~m} \) - Thickness of tank, \( t=0.0127 \mathrm{~m} \) - Buckling factor of safety, \( n=2 \) - Gravitational constant, \( g=9.8 \mathrm{~m} / \mathrm{s}^{2} \) - Density af steel, \( \mathrm{ew}^{-7800 \mathrm{~kg} / \mathrm{m}^{3}} \) - Density of water, \( \theta_{e n}=1000 \mathrm{~kg} / \mathrm{m}^{3} \) For this problem, the anky mode af failure to pratect against is support calumn buckling. According to the Euler column formula, the maximum load that can be supported by a column with ane end free and one end Tasks: (i) Find the tallest water tower design, \( h \). (ii) Find the largest storage capacity design. (iii) Find four other Pareto solutians that are different than the two designs abtained above. (iv) Plat all wour Pareta aptimal designs in the consequence fobjecthel space.

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