midterm2-1

# midterm2-1 - Math 107 Second Midterm Examination...

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Unformatted text preview: Math 107 Second Midterm Examination October 24, 2008 NAME (Please print) Page Score 2 3 4 5 6 7 8 9 Total (Max Possible: 100) Instructions: 1. Do all computations on the examination paper. You may use the backs of the pages if necessary. 2. Put answers inside the boxes (when applicable). 3. Please signify your adherence to the honor code: I, , have neither given nor received aid in completion of this examination. 1 (10 Points) Score Compute the determinant of 4 3 2 1- 2 5- 1- 2 1 2- 2 Using expansion by minors, det A = (- 1) · det 4 2 1- 2- 1- 2- 2 = (- 1) · (- 2) det 4 2- 2- 1 = 2[- 4 + 4] = 0 determinant = 0 2 (10 Points) Score Find a basis for the nullspace of 2 6 2- 4- 8 1- 5 First we will perform Gaussian elimination to find a row echelon form for the matrix: 2 6 2- 4- 8 1- 5 R 3- R 1 / 2 → 2 6 2- 4- 8 = U The first two variables are basic, and the third is free. Setting the free variable x 3 to 1, we solve Ux = 0: 0 =- 4 x 2- 8 x 3 =- 4 x 2- 8 = ⇒ x 2 =- 2 0 = 2 x 1 + 6 x 2 + 2 x 3 = 2 x 1 + 6(- 2) + 2(1) = 2 x 1- 10 = ⇒ x 1 = 5 basis = 5- 2 1 3 (10 Points) Score A lake has a stream flowing in and a stream flowing out at equal rates of 2 cubic meters per minute. The volume of the lake is 10 6 cubic meters. A neighborhood located upstream from the lake has been allowing lawn fertilizer to pollute the lake for several years, leading to a 10% nitrate concentration in the lake. Biologists have determined that in order for the lake to become healthy for fish again, the pollution level has to be reduced to at most 1%. At what rate can the neighborhood discharge nitrates into the stream if the lake is to reach 1% nitrate concentration in 5 years? (Please assume that there are 60*24*365 = 525,600 minutes in a year). Since you are not permitted to use a calculator, you may give your answer as a product of numbers. Let N ( t ) be the volume of nitrate in the lake (in cubic meters), P be the volume concentration of nitrate discharge from the neighborhood (in cubic meters/cubic me- ters). Then N is increased due to inflowing pollution at the pollution concentration P times the volumetric flow rate of the stream in, and N is decreased due to outflowing pollution at volumetric flow rate of the stream out times the lake concentration, which is N ( t ) divided by the lake volume:...
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## This note was uploaded on 08/03/2009 for the course MATH 107 taught by Professor Trangenstein during the Spring '07 term at Duke.

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midterm2-1 - Math 107 Second Midterm Examination...

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