# Mr. Smith decides to feed his pet Doerman pinscher a combination of two dog foods.  Each can of brand A contains 3 units of protein, 1 unit of carbohydrates, and 2 units of fat and costs 60 cents.  Each can of brand B contains 1 unit of protein, 1 unit of carbohydrates, and 4 units of fat and costs 40 cents.  Mr. Smith feels that each day his dog should have at least 6 units of protein, 4 units of carbohydrates, and 12 units of fat.  How many cans of each dog food should he give to his dog each day to provide the minimum requirements at the least cost?

This is a kind of optimization problem, and it can be solved by using a system of inequalities. Let's denote the number of cans of brand A dog food by and the number of cans of brand B dog food by b.  The goal is to minimize the cost,...

This is a kind of optimization problem, and it can be solved by using a system of inequalities. Let's denote the number of cans of brand A dog food by and the number of cans of brand B dog food by b. The goal is to minimize the cost, which can be expressed, in dollars, as

C = 0.6a + 0.4(60 cents for each can of A and 40 cents for each can of B)

The constraints on a and come from the minimum requirements for each nutrient. The amount of protein in cans of brand A dog food and b cans of brand B dog food is

P = 3a + 1b (3 units of protein in each can of A and 1 unit of protein in each can of B). According to Mr. Smith, the amount of protein has to be at least 6 units a day, so

`3a+ b >=6` .

Similarly, the amount of carbohydrates in the same combination of two brands will be

CH = 1a + 1b (1 unit in each can of A and 1 unit in each can of B). The requirement for carbohydrates is

`a+b >=4`

Finally, for fat

F = 2a + 4(2 units in each can of A and 4 units in each can of B). The minimum amount of fat is 12 units:

`2a + 4b >=12`

The three constrains obtained above are graphed below. (a is graphed on the horizontal axis, and is graphed on the vertical axis.) The region formed by the intersecting lines is a triangle with the vertices

(2, 2) (The intersection of 2a + 4b = 12 and a + b = 4)

(1, 3) (The intersection of 3a + b = 6 and a + b = 4)

(1.2, 2.4) (The intersection of 2a + 4b = 12 and  3a + b = 6)

(These results can be verified by solving the corresponding systems of equations.)

The cost (or any function of a and b) will have its maximum and minimum values at the one of the vertices of the triangle. (This is a rule for optimization of a linear function with linear constraints.)

`C(1, 3) = 0.6*1 + 0.4*3 = 1.8`

`C(2,2) = 0.6*2 + 0.4*2 = 2`

`C(1.2, 2.4) = 0.6*1.2 + 0.4*2.4 = 1.68`

The least of these values is 1.68 dollars, which is the cost of 1.2 can of brand A dog food and 2.4 cans of brand B dog food.

So, Mr. Smith should feed his dog 1.2 can of brand A dog food and 2.4 cans of brand B dog food each day. This will cost him \$1.68 a day, which will be the least amount of money he can spend and still satisfy the nutritional requirements.

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