Question

A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply zero units of protein, 2 units of carbohydrates, and 1 unit of fat, and will contain 20 calories. Each ounce of nuts will supply 3 units of protein, 1 unit of carbohydrate, and 2 units of fat, and will contain 30 calories. Every package must provide at least 6 units of protein, at least 10 units of carbohydrates, and no more than 9 units of fat. Find the number of ounces of fruit and number of ounces of nuts that will meet the requirement with the least number of calories. What is the least number of calories?

Solution

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Let

x = ounces of fruit,

y= ounces of nuts.

The problem is to MINIMIZE calories,      z=20x+30yz=20x+30y

subject to:

{3y6... protein 2x+y10... carbohydrates x+2y9... fat \left\{\begin{array}{ll} 3y\geq 6 & \text{... protein } \\ 2x+y\geq 10 & \text{... carbohydrates } \\ x+2y\leq 9 & \text{... fat } \end{array}\right.

and x0,y0.\mathrm{x}\geq 0, \mathrm{y}\geq 0.

Sketch the feasible region in quadrant I (x and y are nonnegative).

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