Calculus help, who's super smart?

Question

This is a crazy word problem, I don't even know where to start.

A large institution is preparing lunch menuse containing foods A and B. Food A contains 1 unit of fat per oz, 2 of carbs per oz, and 1 of protein per oz. Food B contains 1 unit of fat per oz, 1 unit of carbs per oz and 1 unit of protein per oz. Each lunch must provide at least 6 units of fat per serving, no more than 7 units of protein, and at least 10 units of carbs. The institution can purchase food A for $0.12 per oz and food B for $0.08 per oz. How many oz of each food should a serving contain to meet the dietary requirements at the least cost?
Any help would be greatly appreciated, thanks in advance and a quick thanks to all the top contributers out there.

Answer 1

Well, you don't need calculus for this one :)

You can't do it with five ounces at all (not enough fat) and you can't do it with seven ounces (too much protein), so you need six units. To get ten units of carbs requires a minimum of 3 units of food A. Therefore the possibilities are:

3 A, 3 B
4 A, 2 B
5 A, 2 B
6 A, 0 B

Since A is more expensive per unit, 3 A and 3 B is the choice that minimizes cost.

Answer 2

ok, i believe it is a linear programming question.

let x= A, y = B

now, set up the cost function:

P(x, y)= 0.12A + 0.08 B

make a table of the information:
A (column 1) B (column 2)
fat 1 /oz 1 /oz > or = 6
carbs 2 /oz 1 /oz < or = to 7
protein 1 /oz 1 /oz > or = 10

now set up the three equations

A + B > or = 6
2A + B< or = to 7
A + B > or = 10

you re looking for the least cost, so graph these three equations and find the minimum by plugging the value of the points (more than one to choose from) of intersection on the graph into the respective cost function: P(x, y)= 0.12A + 0.08 B

obviously, the smallest P(x, y) you get is the answer.

*remember, x axis = A, y axis = B

hope that was clear enough!

Answer 3

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