91Ó°ÊÓ

Let$\mathit{S}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ be the length of the ${\mathit{t}}^{\mathbf{t}\mathbf{h}}$day of the year$\mathbf{2013}$ in Mumbai (formerly known as Bombay), India (measured in hours, from sunrise to sunset). We are given the following values of$\mathit{S}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ :

$\mathbf{\left[}\begin{array}{cc}\mathbf{t}& \mathbf{S}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\\ \mathbf{47}& \mathbf{11}\mathbf{.5}\\ \mathbf{74}& \mathbf{12}\\ \mathbf{273}& \mathbf{12}\end{array}\mathbf{\right]}$

For example$\mathit{S}\mathbf{\left(}\mathbf{47}\mathbf{\right)}\mathbf{=}\mathbf{11}\mathbf{.5}$, means that the timefrom sunrise to sunset on February16is11hours and30minutes. For locations close to the equator, the function$\mathit{S}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ is well approximated by a trigonometric functionof the form

$\mathit{S}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathit{a}\mathbf{+}\mathit{b}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\frac{\mathbf{2}\mathbf{Ï€}\mathbf{t}}{\mathbf{365}}\mathbf{\right)}\mathbf{+}\mathit{c}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\frac{\mathbf{2}\mathbf{Ï€}\mathbf{t}}{\mathbf{365}}\mathbf{\right)}$

(The period is 365 days, or 1 year.) Find this approximationfor Mumbai, and graph your solution. Accordingto this model, how long is the longest day of the year inMumbai?

Kyle is getting some flowers for Olivia, his Valentine. Being of a precise analytical mind, he plans to spend exactly \(24 on a bunch of exactly two dozen flowers. At the flower market they have lilies (\)3 each), roses (\(2 each), and daisies (\)0.50 each). Kyle knows that Olivia loves lilies; what is he to do?

Question:A lower triangular 3x3 matrix has rank 3 if (and only if) the product of its diagonal entries is nonzero.

Question:A linear system of the form$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\mathbf{0}$ is called homogeneous. Justify the following facts:

a.All homogeneous systems are consistent.

b.A homogeneous system with fewer equations than unknowns has infinitely many solutions.

c.If$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{}\mathbf{and}\mathbf{}\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{2}}}$ are solutions of the homogeneous system$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\mathbf{0}$, then$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{+}\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{2}}}$ is a solution as well.

d.If$\stackrel{\mathbf{→}}{\mathbf{x}}$ is a solution of the homogeneous system$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\mathbf{0}$ andkis an arbitrary constant, then$k\stackrel{→}{x}$ is a solution as well.

Consider a solution$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{1}}}$of the linear system$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}=\stackrel{\mathbf{→}}{b}$. Justify the facts stated in parts (a) and (b):

a. If${\stackrel{\mathbf{→}}{\mathbf{x}}}_{\mathbf{h}}$is a solution of the system$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}=\stackrel{\mathbf{→}}{0}$, then$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{+}\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{h}}}$ is a solution of the system$\mathbf{A}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$.

b. If$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{2}}}$is another solution of the system$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}=\stackrel{\mathbf{→}}{b}$, then$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{+}\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{h}}}$is a solution of the system $\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{+}\stackrel{\mathbf{→}}{0}$.

c. Now suppose A is a$\mathbf{2}\mathbf{×}\mathbf{2}$matrix. A solution vector$\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{1}}}$of the system$\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{+}\stackrel{\mathbf{→}}{b}$is shown in the accompanying figure. We are told that the solutions of the system$\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{0}}$form the line shown in the sketch. Draw the line consisting of all solutions of the system$\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{b}$.

If you are puzzled by the generality of this problem, think about an example first:

$\mathbf{A}\mathbf{=}\left(\begin{array}{l}1â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots 2\\ 3â€\dots â\P Ä\dots â\P Ä\dots â\P Ä\dots 6\end{array}\right)\mathbf{,}\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{=}\left[\begin{array}{l}3\\ 9\end{array}\right]\mathbf{and}\stackrel{\mathbf{→}}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{=}\left[\begin{array}{l}1\\ 1\end{array}\right]$

Consider the equations

$\mathbf{\left|}\begin{array}{c}\mathbf{y}\mathbf{+}\mathbf{2}\mathbf{k}\mathbf{z}\mathbf{=}\mathbf{0}\\ \mathbf{x}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{+}\mathbf{6}\mathbf{z}\mathbf{=}\mathbf{2}\\ \mathbf{k}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{z}\mathbf{=}\mathbf{1}\end{array}\mathbf{\right|}$

where$\mathit{k}$ is an arbitrary constant.

a. For which values of the constant $\mathit{k}$does this system have a unique solution?

b. When is there no solution?

c. When are there infinitely many solutions?

For an arbitrary positive integer$\mathit{n}\mathbf{â‰\yen }\mathbf{3}$, find all solutions ${\mathit{x}}_{\mathbf{1}}\mathbf{,}{\mathit{x}}_{\mathbf{2}}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{,}\mathbf{.}\mathbf{.....}\mathbf{,}{\mathit{x}}_{\mathbf{n}}$of the simultaneous equations ${\mathit{x}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{3}}\mathbf{)}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{4}}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{....}\mathbf{,}{\mathit{x}}_{\mathbf{n}\mathbf{−}\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{n}\mathbf{−}\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{n}}\mathbf{)}$. Note that we are asked to solve the simultaneous equations ${\mathit{x}}_{\mathbf{k}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{k}\mathbf{−}\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{k}\mathbf{+}\mathbf{1}}\mathbf{)}$, for$\mathit{k}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{.}\mathbf{....}\mathbf{,}\mathit{n}\mathbf{−}\mathbf{1}$ .

Make me a crown weighing 60 minae from a mixture of gold, bronze, tin, and wrought iron. Let the gold and bronze together form two-thirds of the weight, the gold and tin together three-fourths, and the gold and iron three-fifths. Tell me how much gold, tin, bronze, and iron you must use. (From the Greek Anthology by Metrodorus, ${\mathbf{6}}^{\mathbf{th}}$century A.D.)