Question

A farmer has 240 acres to plant. He needs to decide how many acres of corn to plant and how many of oats. He can make $40 per acre profit for corn and $30 per acre for oats. However, the corn takes 2 hours of labor per acre to harvest and the oats only take 1 hour per acre. He only has 320 hours of labor he can invest. To maximize his profit, how many acres of each should he plan

Answer 1

Answer:

The acres of each the farmer must plant to obtain the maximum profit is:

• 160 acres of oat.
• 80 acres of corn.

Step-by-step explanation:

First, we're gonna take the data of the exercise:

1. Area: 240 acres.
2. Profit of the corn: $40/acre
3. Profit of the oat: $30/acre
4. Time spent by corn: 2 hour per acre.
5. Time spent by oat: 1 hour per acre.
6. Total working time available: 320 hours.

Now, we're gonna suppose that all the area could be planted with corn (the most profitable), in this form, we obtain the highest profit to take this how reference:

• Total profit with corn = area * profit of the corn
• Total profit with corn = 240 acres * $40/acre
• Total profit with corn = $9600

How you can see, $9600 is the highest profit that the farmer can obtain, but, as each acre of corn needs 2 hours, the farmer would need 480 hours of labor to plant all the area with corn, with the 320 hours availables, the farmer just could plant 160 acres of corn, therefore, the real profit would be:

• Total profit with corn = 160 * $40
• Total profit with corn = $6400

Now, we're gonna make the same exercise but with the oats, to obtain the total profit with this:

• Total profit with oat = area * profit of the oat
• Total profit with oat = 240 acres * $30/acre
• Total profit with oat = $7200

Taking into account the hours of labor availables, the oat gives $800 more profit than the corn, but in that case we'd lose 80 hours of labor that could be used to make a better profit, but, as the oat per hour of labor is more profitable, we're gonna use 2/3 of the area to plant oat and 1/3 to plant corn:

• Area used for oat = 240 acres * 2/3
• Area used for oat = 160 acres
• Area used for corn = 240 acres * 1/3
• Area used for corn = 80 acres

How you can see, with 160 acres of oat, the farmer would need 160 hours of labor and, with 80 acres of corn, the farmer would need 160 hours of labor too, by this, we know all the available time will be used, now, we're gonna calculate the profit in each case:

• Real profit with oat = used area * profit of the oat
• Real profit with oat = 160 acres * $30/acre
• Real profit with oat = $4800
• Real profit with corn = 80 acres * $40/acre
• Real profit with corn = $3200

The total profit to the farmer with this plan is:

• Total profit = Corn profit + Oat profit
• Total profit = $3200 + $4800
• Total profit = $8000

The profit with the given plan is $8000, that is better than the profit only with oat or corn separately and, this is the maximum profit the farmer can make.