Question

The dogs in k-9 kennel must get at least 40 punds of food per day. The food may be a mixture of foods X + Y. The daily diet must include at least 20 units of nutrient A and at least 30 units of nutrient B. The dogs must not get more than 100 pounds of food per day.

A- Food x cost $0.80 per pound and food Y costs $0.40 per pound. What is the least possible cost per day for feeding the dogs?

B- the the price of X is raised to $1.00 per pound, and the price of food Y stays the same, should the kennel change to combination of foods it's using?

Answer 1

@aum @Hero @jdoe0001 @Somy

Answer 2

Let the mixture be made up of "x" pounds of food X and "y" pounds of food Y.
Condition on Nutrient A: x + 1/3 * y >= 20
Condition on Nutrient B: 1/2 * x + y >= 30
Condition on quantity of food: x + y <= 100

Cost of food = 0.8x + 0.4y
Minimize the cost function.

Answer 3

ok so we have our conditions, do we graph each condition and see where or if they intersect?

Answer 4

Graph the first three conditions. Just use the equal sign instead of the inequality sign. Identify the solution space. Note: Since the first two are "greater than or equal", the solution space will be ABOVE and including the line. Since the third inequality is "less than or equal to", the solution space will be BELOW and including that line. Identify all points of intersection. Then test each point in the cost equation and see which one gives you the minimum cost.

Answer 5

http://prntscr.com/4j3djg

Answer 6

The solution space is above the green line, above the purple line, below the orange line, above x-axis and to the right of y-axis.

The points of intersections of these lines are: (0,100), (0,60), (12, 24), (60, 0) and (100, 0). Substitute each point in the cost function 0.8x + 0.4y and see which one gives the minimum cost. And that will be your solution.

Answer 7

ok :) one sec

Answer 8

i got (12,24) as the lowest cost

Answer 9

Yes. 12 pounds of food X and 24 pounds of food Y.
And what is the cost that you are getting when x = 12 and y = 24?

Answer 10

$15.60

Answer 11

cost = 0.8x + 0.4y
At (12,24), cost = 0.8 * 12 + 0.4 * 24 = ?

Answer 12

opps, $19.20

Answer 13

Yes, that is the answer to A. $19.20

Answer 14

is the answer for B to stay the same because it is still the most cost effective?

Answer 15

Now the cost function has changed.
The new cost function = 1.0 *x + 0.4 * y = x + 0.4y

So we need to test each of the earlier 5 points on this new cost function to see which one yields the lowest cost for this new cost function.

Answer 16

Well?