Question

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 batch 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.

Answer 1

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

Hope this helps :)