How many loaves of each type of bread should the baker make to maximize her profit?

Question

Baking a loaf of banana bread takes 2 cups of flour and 1 cup of sugar. A loaf of pumpkin bread takes 3 cups of flour and 2 cups of sugar. A baker has 30 cups of flour and 16 cups of sugar to use. She makes $2.50 profit per loaf of banana bread and $4.00 profit per loaf of pumpkin bread. How many loaves of each type of bread should the baker make to maximize her profit?

Answer 1

I really spent a lot of time at this, trying to find a simple way to explain it.  For one thing, I am new here, and I guess there is no way to show a graph with the feasible region.  That might be good - find a way to explain it without a graph.

 

I am out of time and might visit this and make updates later, but for now here is what I have:

 

Here is what the baker can put together:
15 loaves of bana using up all 30 cups of flour and 15 cups of sugar for each AND no plumpkin because no flour is left for plumpkin. The profit for this would be $2.50 (15) + $4(0) = $37.50
14 loaves of bana using 28 cups of flour and 14 cups of sugar AND 1 still now plumpkin because while there is enough sugar (2 cups), there’s not enough flour left for the plumpkin which needs 3 cups of flour. The profit for this would be $2.50 (14) + $4.00 (0) = $35.00
13 loaves of bana using 26 cups of flour and 13 cups of sugar AND 1 loaf of plumpkin using 3 cups of flour and 2 cups of sugar.
The profit for this would be $2.50 (13) + $4.00 (1) = $36.50.
12 loaves of bana using 24 cups of flour and 12 cups of sugar AND 2 loaves of plumpkin using 6 cups of flour and 4 cups of sugar.
The profit for this would be $2.50 (12) + $4.00 (2) = $38.00 (This uses everything, no wasted left-overs.)
11 loaves of bana using 22 cups of flour and 11 cups of sugar AND 2 loaves of plumpkin using 6 cups of flour and 4 cups of sugar.
The profit for this would be $2.50 (11) + $4.00 (2) = $35.50
10 loaves of bana using 20 cups of flour and 10 cups of sugar AND 3 loaves of plumpkin using 9 cups of flour and 6 cups of sugar.
The profit for this would be $2.50 (10) + $4.00 (3) = $37.00
9 loaves of bana using 18 cups of flour and 9 cups of sugar AND 3 loaves of plumpkin using 9 cups of flour and 6 cups of sugar
Etc.

Let’s make a table:
        x loaves of Bana  y loaves of Plumpkin           x + y loaves of Bana and Plumpkin

Flour   2x                       3y                                  2x + 3y < 30 cups of flour: x loaves use 2 cups ea., y loaves use 3 cups ea.
Sugar 1x                        2y                                  1x + 2y < 16 cups of sugar:x loaves use 3 cups ea., y loaves use 1 cup ea.

The constraints are
2x + 3y = 30 --> x 1 --> 2x + 3y = 30
 x  + 2y = 16 --> x 2 --> 2x + 4y = 32
and contain the maximum value of the linear function. The maximum value occurs at the vertex of the two lines that define the constraints shown above. The vertex is (12,2), meaning 12 loaves of Bana and 2 loaves of Plumpkin will maximize profit.
The profit at (12,2) is
$2.50(12) + $4(2) = $30 + $8 = $38