## Related questions with answers

Suppose that you go into business raising Thoroughbreds and quarter horses. Having studied linear programming, you decide to maximize the feasible profit you can make. Let $x$ be the number of Thoroughbreds, and let $y$ be the number of quarter horses you raise each year.

a. Write inequalities expressing each of the following requirements:

i. Your supplier can get you at most 20 Thoroughbreds and at most 15 quarter horses to raise each year.

ii. You must raise at least 12 horses, total, each year to make the business worthwhile.

iii. A Thoroughbred eats 2 tons of food per year, but a quarter horse eats 6 tons per year. You can handle no more than 96 tons of food per year.

iv. A Thoroughbred requires 1000 hours of training per year. and a quarter horse only 250 hours per year. You have enough personnel to do at most 10,000 hours of training per year.

b. Draw a graph of the feasible region.

c. One of the inequalities has no effect on the feasible region. Which one? Tell what this means in the real world.

d. What is the minimum feasible number of quarter horses?

e. What is the maximum feasible number of Thoroughbreds?

f. Is it feasible to raise no Thoroughbreds? Explain.

g. You can make a profit of $\$ 500$ for each Thoroughbred and $\$ 200$ for each quarter horse. Shade the portion of the feasible region in which the profit would be at least $\$ 5000$ per year.

h. What is the maximum feasible profit you could make per year, and how would you operate in order to attain that profit?

i. How much more profit do you make per year by operating at the optimum point of part $\mathrm{h}$ rather than by operating at the worst feasible point?

Solution

Verifieda)

text${\text{\i}}$) x$\leq$20 and y$\leq$15

ii) x + y $\geq$ 12

iii) 2x + 6y $\leq$ 96

iv)1000x + 250y $\leq$ 10000

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