document.write( "Question 814098: Catherine is making chocolate tea bread and a special oatmeal bread to sell at a bake sale. A batch of chocolate tea bread requires 2 eggs and 2 cups of flour. The oatmeal bread requires 3 cups of flour and 1 egg. Catherine has 12 cups of flour and 8 eggs on hand. She makes a profit of $2 on both kinds of bread. Write a set of constraints that could be used to graph the feasible region. \n" ); document.write( "

Algebra.Com's Answer #490197 by jsmallt9(3758) $\"\"$ $\"About$

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Note: The problem is not clear about the difference, if any, between a loaf of bread and a batch of bread. In the work below I have assumed that a loaf and a batch are the same thing.

\n" ); document.write( "Let's define the following variables:
\n" ); document.write( "c = the number of batches of chocolate tea bread
\n" ); document.write( "m = the number of loaves of oatmeal bread (\"O\" is a bad name for a variable
\n" ); document.write( "Since each batch of chocolate bread requires 2 eggs:
\n" ); document.write( "2c = the number of eggs required for the chocolate bread
\n" ); document.write( "Since each batch of oatmeal bread requires 1 egg:
\n" ); document.write( "1m (or just m) = the number of eggs required for the oatmeal bread
\n" ); document.write( "And...
\n" ); document.write( "2c + m = the total number of eggs needed for both types of bread.

\n" ); document.write( "Using similar logic on the flour we should find that:
\n" ); document.write( "2c + 3m = the total number of cups of flour needed for both types of bread.

\n" ); document.write( "Now for the constraints:
\n" ); document.write( " $\"c+%3E=+0\"$ since we cannot make a negative number of batches
\n" ); document.write( " $\"m+%3E=+0\"$ since we cannot make a negative number of batches
\n" ); document.write( " $\"2c+%2B+m+%3C=+8\"$ since we cannot use more eggs than we have
\n" ); document.write( " $\"2c+%2B+3m+%3C=+12\"$ since we cannot use more flour than we have
\n" ); document.write( "(We could also require that c and m be integers if we believe that fractional batches of bread are impossible.)

\n" ); document.write( "P.S. The objective function would be, since $2 profit is made on each loaf of bread, would be:
\n" ); document.write( "P = 2c + 2m [or 2(c + m)]
\n" ); document.write( "where P is the total profit. \n" ); document.write( "

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