Gilbert is baking cookies and brownies for a bake sale. It costs him $3 for every batch of cookies he makes and $4 for every bat
ch of brownies he makes. Each batch of cookies takes 2 hours to make, and each batch of brownies takes 1 hour to make. Gilbert has budgeted a total of $100 for expenses and has allowed himself 45 hours to make the cookies and brownies. He makes a profit of $5 for each batch of cookies, and he makes a profit of $4.50 for each batch of brownies. The system of inequalities that represents this situation, (3x+4y≤100) (2x+y≤45), is shown graphed in the first quadrant with the number of batches of cookies along the x-axis and the number of batches of brownies along the y-axis.
The equation p=5x+4.5y represents the profit (p) that Gilbert makes if he sells all his baked goods.
What combination of baked goods allows Gilbert to maximize his profit?
A.If Gilbert makes 0 batches of cookies and 45 batches of brownies, he will make a maximum profit of $202.50.
B.If Gilbert makes 0 batches of cookies and 25 batches of brownies, he will make a maximum profit of $112.50.
C.If Gilbert makes 32 batches of cookies and 0 batches of brownies, he will make a maximum profit of $160.00.
D.If Gilbert makes 22 batches of cookies and 0 batches of brownies, he will make a maximum profit of $110.00.
E.If Gilbert makes 20 batches of cookies and 4 batches of brownies, he will make a maximum profit of $118.00.
F.If Gilbert makes 16 batches of cookies and 13 batches of brownies, he will make a maximum profit of $138.50.
Let c=the number of batches for the cookies; b=that for the brownies. If $P=the profit, then maximize P=5c+4.5b, subject to the constaints: 3c+4b<=100 (cost) 2c+b<=45 (time) b,c >=0 The simplest way to find the suitable b & c is to solve 3c+4b=100 2c+b=45 for b & c The result is b=13 & c=16 => max. p=5(16)+4.5(13)=$138.5
