\documentclass[12pt]{article}\usepackage{ammaths}\begin{document}


\en
\partie{Homework \#1}

\setcounter{probpart}{0}

\begin{center}
\large\bf \textsf{The CPL composition of an aliment}
\end{center}

When analysing the quality of our food, a first approch is the composition in carbohydrates, proteins, lipids. These are
the three main families of nutrients in an aliment : the carbohydrates (sugars) are a source of energy, the proteins
(containing nitrogen) help build cells and muscles, the lipids (fat) store energy. 

Each aliment is made of these three kinds of nutrients, with different percentages. For example, spinach is made of
$55\:\%$ carbohydrates, $40\:\%$ proteins and $5\:\%$ lipids.

\probpart{-- The aliments map}

In this part, we will establish an aliments map, a graph were aliments will be represented by points. As an example the
point $S$ representing spinach has been placed. We will say that the coordinates of this point $S$ are $(55,40,5)$. 

To read the CPL coordinates of a point (or an aliment), lines parallel to the three axes must be drawn through this
point. This construction has been done for the point $S$ on the map. On this map also appear the points $M$, $C$
, $F$ and $P$ representing milk, carrot, flour and fish.

%\begin{multicols}{2} 
\begin{center}
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	%
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	%
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	%
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	\psline[linewidth=.1pt](6,0)(8,3.47)
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	\psline[linewidth=.1pt](6,0)(3,5.2)
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	%
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	\psline[linewidth=.05pt](1,0)(.5,.87)
	\psline[linewidth=.05pt](3,0)(1.5,2.6)
	\psline[linewidth=.05pt](5,0)(2.5,4.33)
	\psline[linewidth=.05pt](7,0)(3.5,6.07)
	\psline[linewidth=.05pt](9,0)(4.5,7.8)
	\psline[linewidth=.05pt](.5,.87)(9.5,.87)
	\psline[linewidth=.05pt](1.5,2.6)(8.5,2.6)
	\psline[linewidth=.05pt](2.5,4.33)(7.5,4.33)
	\psline[linewidth=.05pt](3.5,6.07)(6.5,6.07)
	\psline[linewidth=.05pt](4.5,7.8)(5.5,7.8)
	%
	\psline[linewidth=.5pt,linestyle=dashed]{->}(2.49,3.47)(2,3.47)
	\psline[linewidth=.5pt,linestyle=dashed]{->}(2.49,3.47)(5.25,8.23)
	\psline[linewidth=.5pt,linestyle=dashed]{->}(2.49,3.47)(4.5,0)
	%
	\psdots[dotstyle=*,dotscale=1]%
	(2.49,3.47)(1,.87)(5,1.73)(.75,1.3)(6.25,6.5)
	\uput{.2}[30](2.49,3.47){$S$} \uput{.2}[270](1,.87){$C$}
	\uput{.2}[35](5,1.73){$M$} \uput{.2}[180](.75,1.3){$F$}
	\uput{.2}[0](6.25,6.5){$P$}
	%
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	\rput(7.9,-.5){20} \rput(.6,1.73){20} %\rput(1.6,3.47){40}
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	\rput(7.4,5.2){40} \rput(8.4,3.47){60} \rput(9.4,1.73){80}
	\rput(1.5,-1){Carbohydrates $\%$}
	\rput{60}(3,6.93){Proteins $\%$}
	\rput{300}(9.8,2){Lipids $\%$}
     \end{pspicture}\\
{\sl CPL coordinates of an aliment}
\end{center}

%\pagebreak
\begin{enumerate}
\item Give the CPL composition of milk, carrot, flour and fish.
\item Build an equaliteral triangle with side 10cm and reproduce the aliments map. Place on it the points $E$, $H$, $B$
and $A$ representing egg, sugar, butter and meat, according to the table below.
\begin{center}
     \begin{tabular}{|l|c|c|c|l|c|c|c|}
	\hline
	\multicolumn{8}{|c|}{\sc A few CPL compositions} \\ \hline\hline
	 &C&P&L&&C&P&L\\
	\hline
	Rice & 90 & 10 & 0 & Green beans & 75 & 25 & 0 \\
	\hline
	Biscuits & 80 & 10 & 10 & Meat & 0 & 50 & 50 \\
	\hline
	Apple pie & 75 & 5 & 20 & Flour & & & \\
	\hline
	Chocolate & 70 & 5 & 25 & Jam & 98 & 2 & 0 \\
	\hline
	Pasta & 90 & 10 & 0 & Butter and oil & 0 & 0 & 100 \\
	\hline
	Egg & 0 & 40 & 60 & Spinach & 55 & 40 & 5 \\
	\hline
	Sugar & 100 & 0 & 0 & Fish & & & \\
	\hline
	Potatoes & 90 & 10 & 0 & Apple & 95 & 5 & 0 \\
	\hline
	Bread & 90 & 10 & 0 & Milk & & & \\
	\hline
	Carrot & & & & Honey & 100 & 0 & 0 \\
	\hline
	Cheese & 0 & 50 & 50 & Cauliflower & 70 & 25 & 5 \\
	\hline
	Lentils & 75 & 25 & 0 & Cabbage & 60 & 35 & 5 \\
	\hline
     \end{tabular}
\end{center}

\item Place the points $U$ (cauliflower) and $K$ (chocolate) on the map. Where, on the map can you find the aliments with $70\%$
carbohydrates ?
\item Where can you find the aliments with the following elements ?
%\vspace*{-10pt}
\begin{multicols}{3}
	 \begin{enumerate}
	    \item $50\:\%$ carbohydrates ;
	    \item $10\:\%$ proteins ;
	    \item $25\:\%$ lipids.
	 \end{enumerate}
     \end{multicols}
\end{enumerate}
%\end{multicols}

\probpart{-- The ideal region}

We note $C$, $P$ and $L$ the proportions in pourcentages of an aliment. For the following questions, you will crosshatch
on the map the rejected zone.
\begin{enumerate}
\item Where are the aliments such that
	    %\vspace*{-10pt}
	    \begin{multicols}{3}
		\begin{enumerate}
	    \item $C > 50$ ?
	    \item $P > 10$ ?
	    \item $L > 25$ ?
		\end{enumerate}
	    \end{multicols}
    %\vspace*{-12pt}
\item Dietitians have established that the proportions best stuited for human beings are :
     $$\left\lbrace \begin{array}{c}
	    50  <  C  <  60 \\
	    10  <  P  <  20 \\
	    25  <  L  <  35\\
	\end{array}
	\right.$$
     Find the adequate region on the map.
\item Which of the aliments in the above map are in this region ?
\end{enumerate}

\probpart{-- A few meals}


Any meal should be conceived such that, by mixing up aliments, the proportions C, P, L are in the ideal region.

\begin{enumerate}
\item \emph{Fast-food restaurants}\\
	The standard meal in a fast-food restaurant is made of a hamburger (35g of bread, 35g of meat), a portion of
fries (90g of potatoes and 10g of oil) and a carbonated soft drink (20g of sugar).
	\begin{enumerate}
	\item Give in grams the CPL composition of 35g of bread.
	\item Give, first in grams then in percentages, the CPL composition of a standard fast-food meal.
	\item Place on the map the point $F$ representing this meal. Is it in the ideal region ?
	\end{enumerate}
\item \emph{The pound cake}\\
The recipe of the \emph{pound cake} is very simple : one fourth sugar, one fourth flour, one fourth butter, one fourth
eggs (the fourth are counted over the mass of each ingredient).
	\begin{enumerate}
     \item What are the CPL coordinates of the point $Q$ representing the pound cake ?
     \item The point $Q$ is not in the ideal region : change the recipe so that it is in it.
	\end{enumerate}
\end{enumerate}


\end{document}
