Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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The following picture represents a network of interconnected pipelines. The arrows represent the flow measured by ft³ per minutes. The values above the arrows represent the flow in the given segment, the flow is regulated by the valves at the intersections. Of course the amount of water per minute entering each connection, has to be equal to the amount of water exiting the connection. This leads to five equations. 10 20 40 6 Write down the system of linear equations that represents the flow at each relevant valve. 7 Find the augmented matrix and its row reduced echelon form. 8 How many solutions are there? Notice that the valves allow only one direction for the flow. This means that we require each r; not to be negative. A solution with all positive r; is called feasible. 9 Express all the feasible solutions in terms of inequalities. Suppose you will need to repair the pipe between connections A and B, you will want to have 10 the flow between A and B as minimal as possible. What is the minimal value of æg to have a feasible solution. 11 Using what you found in the previous answer write all the values of ri, for i = 1, ..., 6 and make sure they are a solution of your system. What happens if you reverse the flow in the segment BD (by substituting the valve)? Find the 12 minimal value of r6 that gives you a feasible solution.
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Transcribed Image Text:The following picture represents a network of interconnected pipelines. The arrows represent the flow measured by ft³ per minutes. The values above the arrows represent the flow in the given segment, the flow is regulated by the valves at the intersections. Of course the amount of water per minute entering each connection, has to be equal to the amount of water exiting the connection. This leads to five equations. 10 20 40 6 Write down the system of linear equations that represents the flow at each relevant valve. 7 Find the augmented matrix and its row reduced echelon form. 8 How many solutions are there? Notice that the valves allow only one direction for the flow. This means that we require each r; not to be negative. A solution with all positive r; is called feasible. 9 Express all the feasible solutions in terms of inequalities. Suppose you will need to repair the pipe between connections A and B, you will want to have 10 the flow between A and B as minimal as possible. What is the minimal value of æg to have a feasible solution. 11 Using what you found in the previous answer write all the values of ri, for i = 1, ..., 6 and make sure they are a solution of your system. What happens if you reverse the flow in the segment BD (by substituting the valve)? Find the 12 minimal value of r6 that gives you a feasible solution.
You probably have seen the formula E- i = }n² + }n This means that if you want to know the sum of the first 100 natural number, this number is 1 +-100 1+2+3+ ...+ 100 = It is told that this formula was proven by Gauss at the age of 5, when the kindergarten teacher gave her students this problem, tired of the noise that they were making. Turns out, there are infinitely many of these formulas. If you want to know the value of the sum of the first n numbers raised to the k-th power ¿* = ao + a¡n + . .+ ak+1nk+1 ik For example: 1 1 E² = 1+22 + 32 + · . + n² = +교 + 공 i=1 Notice that when you want to sum the squares of numbers, the formula will yield a polynomial of degree 3 and more in general if you sum the kth power of numbers you will get a polynomial of degree k+1. The goal is to find the formula for k = 6. 1. Specialize the formula in (*) for k = 6. 2. What are the variable you want to solve for? 3. How many equations do you need? 4. Find a system of 8 different equations. 5. Solve the system with Matlab and give the formula.
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Transcribed Image Text:You probably have seen the formula E- i = }n² + }n This means that if you want to know the sum of the first 100 natural number, this number is 1 +-100 1+2+3+ ...+ 100 = It is told that this formula was proven by Gauss at the age of 5, when the kindergarten teacher gave her students this problem, tired of the noise that they were making. Turns out, there are infinitely many of these formulas. If you want to know the value of the sum of the first n numbers raised to the k-th power ¿* = ao + a¡n + . .+ ak+1nk+1 ik For example: 1 1 E² = 1+22 + 32 + · . + n² = +교 + 공 i=1 Notice that when you want to sum the squares of numbers, the formula will yield a polynomial of degree 3 and more in general if you sum the kth power of numbers you will get a polynomial of degree k+1. The goal is to find the formula for k = 6. 1. Specialize the formula in (*) for k = 6. 2. What are the variable you want to solve for? 3. How many equations do you need? 4. Find a system of 8 different equations. 5. Solve the system with Matlab and give the formula.
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