Math 125 Fall 2011 Exam 2 - Solutions

Transcription

1 Math 5 Fall 0 Exam - Solutions. (0 points) Suppose an economy consists of three industries: natural gas, oil, and electricity. For each $ of natural gas produced, $0.0 of natural gas is consumed, $0.0 of oil is consumed, and $0.0 of electricity is consumed. For each $ of oil produced, $0.0 of natural gas is consumed, $0.50 of oil is consumed, and $0.0 of electricity is consumed. For each $ of electricity produced, $0.50 of natural gas is consumed, $0.0 of oil is consumed, and $0.70 of electricity is consumed. The demand for natural gas is $700 million, for oil is $400 million, and for electricity is $00 million. (a) Find the consumption matrix C. Label all rows and columns. ng oil elec...5 ng raw usage C =..5. oil raw usage...7 elec raw usage (b) Find the demand vector D. Label the entries. 700 D = 400 million $ 00 ng oil elec (c) Is this economy productive? Justify your answer no justification, no points. If the economy is productive, find the production schedule x that meets the demand be sure to write down an explicit formula for x before you solve for it and to label the entries of x after you solve for it. Row sums are =.9 <, =.9 <, =.9 < and so the economy os productive. Then x = (I C) D = = = = ng oil elec To meet the demand, $5750 millions in nat gas, $5750 million inoil, and $7500 million in electricity must be produced.

2 . (5 points) Given v = 5 4, v = 7 4, v = 9, v 4 = 9 for each of the statements below, circle the correct answer: True or False. (a) True/ False : The vectors v and v 4 are orthogonal. Show that v v 4 = = 4 0 so non-orthogonal. (b) (c) True /False: The vectors v and v are parallel. v = v so yes, parallel. True /False: The vectors v and v 4 are orthogonal. v v 4 = = 0 (d) True /False: The vector v is a linear combination of the vectors v and v. v = v (e) True/ False : Two of the vectors above have the same length. Check each length!. (5 points) For each of the statements below, circle the correct answer: True or False. (a) True /False: If A = I, then an inverse of A is A. (A )A = A(A ) = A = I so apply the definition of inverse of a matrix. (b) True/ False : If A = O (the zero matrix), then A = O. [ ] This is only true if A is invertible. The result is true, for example, if A = (c) True/ False : If A is invertible, then the inverse of A is A. The inverse of A is A, i.e. (A)( A ) = I (d) True /False: If A is invertible, then its RREF is I. Yes, this was a proposution in class. (e) True /False: If A and B are each 4 4 matrices of row rank 4, then so is AB. Row operations take A to I, so those same row operations take AB to B. Now apply the row operations as well that take B to I. Thus row operations take AB to I. I 4 has row rank (0 points) The Rain Forest Juice Company is a tiny start-up company that sells blends of fruit juice, which are made from organic guava juice, organic mango juice, and Evian spring water. Rain Forest has 840 fluid ounces of guava juice, 450 fluid ounces of mango juice, and 60 fluid ounces of water. They have three different juice boxes: Refreshing, Super Exotic, and Light. Refreshing is made from fluid ounces of guava juice, fluid ounces of mango juice, and fluid ounce of water. Super Exotic is made from 0 fluid ounces of guava juice, fluid ounces of mango juice, and fluid ounce of water. Light is made from fluid ounces of guava juice, fluid ounces of mango juice, and fluid ounce of water. Refreshing sells for 80 a box. Super Exotic is on sale and sells for 60 a box. And Light sells for 50 a box.

3 (a) Set up the linear program to maximize income (i.e. what inequalities must be satisfied, what function is to be optimized?). Labeling of variables and equations is important. maximize.8x +.6x +.5x dollars x = boxes Refreshing subject to x + 0x + x 840 guava - oz x = boxes Super E x + x + x 450 mango - oz x = boxes Light x + x + x 60 water - oz x 0, x 0, x 0 (b) Write down the initial simplex table corresponding to the linear program. Label the columns and rows. objective guava mango water z x x x x 4 x 5 x ratios: 840/ = / = / = 60 (c) Find the maximum income using the simplex algorithm, provided the algorithm does not fail. If the simplex algorithm fails, give a full explanation. If the algorithm does not fail, write down the maximum income, how many boxes of each type of juice blend is needed to achieve this maximum, and how many fluid ounces of each of the ingredients are left over, if any. Whether the simplex algorithm fails or not, write down all possible simplex tables (i.e. all intermediate tables and a final one, if it exists) ratios: 5/.5 = /.5 = 6 Thus, the maximum profit is $96.0 and it is achieved by making 44 boxes of Refreshing, 0 boxes of Super E and 6 boxes of Light. 5. (0 points) Given the vectors v =, v =, b = find all real numbers a for which the vector b is a linear combination of the vectors v and v. For each a that you find, find the scalars r and s for which b = r v + s v. Write out all your steps, indicating the row operations performed and including a new matrix for each row a 0

4 operation. r s a 0 a 0 5 a a a 0 a/ a a 0 a/ a R = R R R = R + R R = R /5 R = R 5R We must have + a = 0 for consistency, so set a = to get 0 / /5 R = R R 0 / and so r = 5, s = (6 points) Suppose a, b, c, d are nonzero real numbers. Let a 0 c d d 0 b a A = 0 b c d B = 0 c b a a b 0 d d c 0 a (a) Which of the following is the (, )-entry of B A? Circle one. A. c b B. c + b C. b c D. b + c b a = b ( c) = b + c (b) Which of the following is the (4, )-entry of AB t? Circle one. A. a B. d C. 0 D. Cannot be found There are only three rows in each matrix! 4

5 (c) If A = [ a 6 a a + 6 which of the following choices for a makes A invertible? Circle one. A. Any real number except 6 B. Any real number except and C. 6 D. Any real number except - and 6 ] det A = ( a+6) a(a 6) = (a 6)(+a) = 0 implies a = 6 or a =. We must avoid both. 7. (4 points) Let l denote the line in R passing through P = (,, ) and P = (4,, ). For each of the statements below, circle the correct answer: True or False. (a) True/ False : A vector equation of the line l is x = + t, y = + t, z = t This is in component form, not vector form. (b) True/ False : A vector equation of the line is x = (,, ) + t(4,, ) (c) The direction should be (4,, ) (,, ) = (,, ) True /False: A vector equation of the line is x = (4,, ) + t(,, ) See solution for part b). (d) True/ False : The point (7, 4, ) does not lie on the line. (7, 4, ) = (4,, )+t(,, ) t(,, ) = (7, 4, ) (4,, ) = (,, ) t =. 5