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A bakery is making whole-wheat bread and apple bran muffins. For each batch of break they make $35 profit. For each batch of muffins, they make $10 profit. The break takes 4 hours to prepare and 1 hour to back. The muffins take 0.5 hours to prepare and 0.5 hours to bake. The maximum preparation time available is 16 hours. The maximum bake time available is 10 hours. Let x = # of the batches of bread and y = # of batches of muffins. Outline the feasible region that can be used to find the number of batches of bread and muffins that should be made to maximize profits? Use the color RED to indicate the feasible region!

Sagot :

shreejitadhakal

Step-by-step explanation:

let the number of wholewheat bread batches be x

let the number of muffin batches be y

prep condition: 4x + (1/2)y ≤ 16 or 8x + y ≤ 32

baking condition : 1x + (1/2)y ≤ 10 or 2x + y ≤ 20

graph each of those in the first quadrant shading in the region below each one.

Final shading is the region satisfying both conditions.

profit = 35x + 10y

allow this line to "slide" away from the origin as far as you can.

It should be clear that the farthest we can go is the intersection of

8x+y = 32 and

2x + y = 20 or the x and y intercepts of our two boundary lines.

subtract them as they are:

6x = 12

x = 2

then in our head , y = 16

max profit = 2(35) + 10(16) = 230

Just to make sure, I will test the intercepts of our intersecting region, namely (4,0) and (0,20)

for (4,0) profit = 4(35)+1 = 140

for (0,32) profit = 0 + 10(20) = 200

so max profit is 230 when x=2 and y=16

(notice all the prep time and baking time is utelized)

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