document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186926>Question 1186926</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In order to ensure optimal health, a lab technician needs to feed the rabbits a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protein. But the rabbits should be fed no more than five ounces of food a day.\r<BR>\n" );
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document.write( "Rather than order rabbit food that is custom-blended, it may be cheaper to order Rabbit-Gro and Lucky-Rabbit, and blend them for an optimal mix. Rabbit-Gro contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per ounce, and costs $0.20 per ounce. Lucky-Rabbit contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per ounce, at a cost of $0.30 per ounce.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849328 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849328\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To determine the most cost-effective mix of Rabbit-Gro and Lucky-Rabbit food that meets the rabbits' nutritional needs, we can use a system of inequalities and solve for the optimal amounts of each food.\r<BR>\n" );
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document.write( "**Let:**\r<BR>\n" );
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document.write( "*   x = ounces of Rabbit-Gro<BR>\n" );
document.write( "*   y = ounces of Lucky-Rabbit\r<BR>\n" );
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document.write( "**Objective:** Minimize cost: C = 0.20x + 0.30y\r<BR>\n" );
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document.write( "**Constraints:**\r<BR>\n" );
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document.write( "*   **Fat:** 8x + 12y ≥ 24<BR>\n" );
document.write( "*   **Carbohydrates:** 12x + 12y ≥ 36<BR>\n" );
document.write( "*   **Protein:** 2x + y ≥ 4<BR>\n" );
document.write( "*   **Total Food:** x + y ≤ 5<BR>\n" );
document.write( "*   **Non-negativity:** x ≥ 0, y ≥ 0\r<BR>\n" );
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document.write( "**Solving the Problem:**\r<BR>\n" );
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document.write( "One way to solve this is graphically. Plot each inequality on a graph with x and y axes. The feasible region is where all inequalities are satisfied. The optimal solution will be at one of the vertices (corners) of this region.\r<BR>\n" );
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document.write( "Alternatively, you can use linear programming techniques (like the simplex method or software tools) to find the optimal solution.\r<BR>\n" );
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document.write( "**Graphical Method (Sketch):**\r<BR>\n" );
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document.write( "1.  **Plot the lines:** Treat each inequality as an equation and plot the lines on a graph.<BR>\n" );
document.write( "2.  **Shade the regions:** Shade the appropriate side of each line based on the inequality sign.<BR>\n" );
document.write( "3.  **Identify the feasible region:** The feasible region is where all shaded areas overlap.<BR>\n" );
document.write( "4.  **Find the vertices:** Determine the coordinates of the vertices of the feasible region.<BR>\n" );
document.write( "5.  **Evaluate the objective function:** Plug the x and y coordinates of each vertex into the cost equation (C = 0.20x + 0.30y).<BR>\n" );
document.write( "6.  **Optimal solution:** The vertex that yields the lowest cost is the optimal solution.\r<BR>\n" );
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document.write( "**Linear Programming:**\r<BR>\n" );
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document.write( "Linear programming is a more efficient way to solve this type of optimization problem, especially if there are many variables or constraints.\r<BR>\n" );
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document.write( "**Expected Outcome:**\r<BR>\n" );
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document.write( "By solving the system of inequalities and minimizing the cost function, you'll find the optimal number of ounces of Rabbit-Gro and Lucky-Rabbit to feed the rabbits while meeting their nutritional requirements and staying within the 5-ounce limit.\r<BR>\n" );
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document.write( "*The solution will likely involve a combination of both foods, but it's possible that one food alone might be the most cost-effective option.*<BR>\n" );
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