Question

Your small farm encompasses 120 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 $3,000 for each acre of tomatoes, $1,400 for each acre of lettuce and $400 for each acre of carrots. You would like to spend at least $480 on fertilizer and your farm laborers can supply up to 600 hours per week. How many acres of each crop should you plant to maximize total profits?
tomatoes acre(s) =
lettuce acre(s) =
carrots acre(s)=
profit $ =

In this event, will you be using all 120 acres of your farm?

Answer 1

Answer:

• tomatoes acre(s) = 120
• lettuce acre(s) = 0
• carrots acre(s)= 0
• profit $ = $360,000
• all 120 acres are used

Step-by-step explanation:

You can write the linear system model as follows. Let t, l, c represent acres of tomatoes, lettuce, and carrots, respectively. The we have ...

  t + l + c ≤ 120 . . . . . constraint on available land

  5t +4l +2c ≥ 480 . . requirement for spending on fertilizer

  4t +2l +2c ≤ 600 . . constraint on available labor

  t ≥ 0; l ≥ 0; c ≥ 0 . .  requirement for non-negative acres

Then the objective function (profit) is ...

  p = 3000t +1400l +400c

The linear programming problem is to maximize p subject to the above constraints.

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Any of a variety of solvers can find the solution to this problem. That solution is ...

  (t, l, c) = (120, 0, 0) and p = 360,000

In summary ...

  tomatoes acre(s) = 120 (all available acres are used)

  lettuce acre(s) = 0

  carrots acre(s) = 0

  profit $ = $360,000

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Additional comments

This solution suggests that it can be found simply by examining the profit associated with each unit of resource.

  profit per acre is maximized for tomatoes, at $3000 per acre

  profit per fertilizer dollar is maximized for tomatoes, at $600 per dollar

  profit per labor hour is maximized for tomatoes, at $750 per hour

That is, the profit per acre is maximized for tomatoes, regardless of the resource being considered. Thus it make sense to put all of the acreage in tomatoes. At $5 per acre for fertilizer, we use $600 worth of fertilizer. At 4 hours per week per acre, we use 480 hours of labor, so not all available labor is used.