Alg1131W10T3 - Applications

Sunday, 13 May 2018

11:53 PM

Machine generated alternative text: The technique for solving systems of linear equations goes back to the ancient Chinese text Nine Chapters of the Mathematical Alts. When we apply this technology in specific situations, a first step is to identify the variables that make up the unknowns; once we realize that a system of equations is involved. It is often a good idea to explicitly identify which variable names correspond to which real-life variables in a problem. Let's try to find a parabola of the form y = ac2 + bx + c which passes through the three points [1, 4], [2, 9] and [—1, 6]. Naively we expect this to be possible, since there are three degrees of freedom in the problem, namely choosing a, b and c, and there are three constraints. If we plug the three points into the parabola; we get three equations We recognize that we have O a system of linear equations in and y @ a system of linear equations in a, b and c O a system of quadratic equations in x and y O a system of quadratic equations in a, b and c O and b is the vector Note: • The Maple Syntax for the matrix • The Maple Syntax for the vector 1 4 7 10 11 12 2 5 8 3 6 9 11213>, < 41516 > , < 71819 is < 10, 11, 12 > If we fully row-reduce this matrix, we obtain (U I c) where U 1 o o 1 o o o 1 and c is the vector O This means that the polynomial we need is y Note: the Maple syntax for the polynomial 13x2 — 14T — 15 is

 

Machine generated alternative text: Suppose we have a neuork of roads and car rental customers over 5 possible routes, as in the video. Next week, five new customers take short trips of lengths a + b we can deduce that 14 and b + d 5. From this

 

Machine generated alternative text: To keen her mathematical skills, the new manager intends to check the previous calculations by using the total milages of five new customers a b If these customers took the routes ecab becabd ecae abde then which of the following matrices could the manager use to set up a linear system involving the five lengths? cdbe o 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 o 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 If the new manager row reduces the correct matrix above; she will find that she can get it into the row echelon form 1 o o o o o 1 o o o o o 1 o o 0 0 o 1 0 y 1 1 w where x The correct conclusion is that in general: O and w O O she can determine the lengths of all five highways she can determine none of the lengths of the highways she can determine the lengths of all the highways except one.

 

Machine generated alternative text: While reading your grocery receipts from last week, you discover that you purchased: • one apple and one banana for $3.25 • one banana and one pear for $2.5 • one pear and one apple for $2.75. Can you determine the individual prices of apples; bananas and pears? @ Of course O Maybe O Not a chance O This is a fruitless activity. Possible dollar values of these fruit are: • apples cost $ 1 75 • bananas cost $ 1.50 • pears cost $ 1 O each O each O each.

 

Machine generated alternative text: Upon investigating your grocery receipts for a different week, you discover that you purchased: • one apple and one banana for 2.0 dollars • one banana and one pear for 3.25 dollars • one pear and one persimmon for 4.50 dollars • one persimmon and one apple for 3.25 dollars. Can you determine the individual pnce of apples; bananas, pears and persimmons? O of course O maybe @ not a chance. Would your answer change if the receipt amounts were different? @ no.

 

 

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