document.write( "Question 1202521: The top two dolls that a toy manufacturer makes are called Baby Wiggles and Sleepy Baby.To makes a case of Baby Wiggles takes 5 units of raw material and 1 unit of time to assemble.To make a case pf Sleepy Baby takes 3 units of raw material and 2 units of time to assemble.On a given day the manufacturer has at most 150 units of raw material and 44 units of time.If the manufacturer makes a profit of RM120 on each case of Baby Wiggles and RM100 on each case of Sleepy Baby,how many cases of each type of doll should the manufacturer make in order to maximize profit? \n" ); document.write( "

Algebra.Com's Answer #837412 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Using a graphing calculator to do most of the work for you can get you to the answer faster -- if you know how to use the graphing calculator. But in finding the answer that way you don't get much practice in doing math.

\n" ); document.write( "The following is my preferred method for demonstrating to a student the process for solving this kind of problem.

\n" ); document.write( "x = # of cases of Baby Wiggles
\n" ); document.write( "y = # of cases of Sleepy Baby

\n" ); document.write( "The constraints based on availability of raw materials and assembly time are these:

\n" ); document.write( " $\"5x%2B3y%3C=150\"$
\n" ); document.write( " $\"x%2B2y%3C=44\"$

\n" ); document.write( "The other two constraints state the obvious fact that the two numbers x and y can't be negative:

\n" ); document.write( " $\"x%3E=0\"$
\n" ); document.write( " $\"y%3E=0\"$

\n" ); document.write( "Step 1: Graph the constraint boundary lines to determine the feasibility region.

\n" ); document.write( "For reasons you will see in a bit, I recommend using slope-intercept form.

\n" ); document.write( " $\"y%3C=%28-5%2F3%29x%2B50\"$
\n" ); document.write( " $\"y%3C=%28-1%2F2%29x%2B22\"$

\n" ); document.write( "The inequalities in this form show that the solutions must lie on or below both constraint boundary lines.

\n" ); document.write( "Step 2: Solve the pair of equations for the constraint boundary lines to find their point of intersection and thus determine the corner points of the feasibility region.

\n" ); document.write( "I leave it to you to do the relatively easy work to find that the corners are (0,0), (0,22), (24,10), and (30,0).

\n" ); document.write( "Step 3: Find the corner point where the profit (objective function) is maximized.

\n" ); document.write( "(Note it is NOT necessary, as nearly all references say, to evaluate the objective function at all corners of the feasibility region. That corner point can be determined from the slopes of the objective function and the constraint boundary lines.)

\n" ); document.write( "The objective (profit) function is

\n" ); document.write( " $\"P=120x%2B100y\"$

\n" ); document.write( "Put this function in slope-intercept form:

\n" ); document.write( " $\"100y=-120x%2BP\"$
\n" ); document.write( " $\"y=%28-6%2F5%29x%2BC\"$ for some number C

\n" ); document.write( "The profit function will be maximized when a line with slope -6/5 just touches the feasibility region. Since the slope of the objective function, -6/5, is between the slopes of the two constraint boundary lines, the profit function will be maximized at the intersection of the two constraint boundary lines.

\n" ); document.write( "So the profit is maximized when x = 24 and y = 10.

\n" ); document.write( "ANSWER: The profit is maximum when 24 cases of Baby Wiggles and 10 cases of Sleepy Baby are produced.

\n" ); document.write( "CHECK:

\r\n" );
document.write( "  corner    objective function value, 120x+100y\r\n" );
document.write( " -----------------------------------------------\r\n" );
document.write( "   (0,0)    0+0= 0\r\n" );
document.write( "   (0,22)   0+2200 = 2200\r\n" );
document.write( "   (24,10)  2880+1000= 3880\r\n" );
document.write( "   (30,0)   3600+0 = 3600

\n" ); document.write( " \n" ); document.write( "

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