Your small farm encompasses 140 acres, and you are planning to grow tomatoes, le

Question

Your small farm encompasses 140 acres, and you are planning to grow tomatoes, lettuce, and carrots in the coming planting season. Fertilizer costs per acre are: $5 for tomatoes, $4 for lettuce, and $2 for carrots. Based on past experience, you estimate that each acre of tomatoes will require an average of 4 hours of labor per week, while tending to lettuce and carrots will each require an average of 2 hours per week. You estimate a profit of $2,000 for each acre of tomatoes, $1,500 for each acre of lettuce and $500 for each acre of carrots. You would like to spend at least $560 on fertilizer (your niece owns the company that manufactures it) and your farm laborers can supply up to 700 hours per week. How many acres of each crop should you plant to maximize total profits? HINT [See Example 3.] acre(s) tomatoes acre(s) lettuce acre(s) Carrots profit In this event, will you be using a 140 acres of your farm? Yes No

Explanation / Answer

Let the variables be;

T, L C;

Maximize Profit = 2000T + 1500L+500C;

subject to conditions:

5T + 4L + 2C<=560;

4T+2L+2C<=700;

Using simplex method; we get

The following table;

Table #1
t l c s1 s2 p   
5 4 2 1 0 0 560
0 2 2 0 1 0 700
-2000 -1500 -500 0 0 1 0

Table #2
t l c s1 s2 p   
1 0.8 0.4 0.2 0 0 112
0 2 2 0 1 0 700
0 100 300 400 0 1 224000

Optimum solution is:

T=112acres;L=0;C=0;

P = $224000;

No we are not using full farm;

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