Your farm encompasses 400 acres, and you are planning to grow soy beans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: $5 for soy beans, $2 for corn, and $1 for wheat.You estimate that each acre of soy beans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of $3,000 for each acre of soy beans, $2,000 for each acre of corn, and $1,000 for each acre of wheat. You can afford to spend no more than $2,400 on fertilizer, and your farm laborers can supply 2,800 hours per week. How many acres of each crop should you plant to maximize total profits In this event, will you be using all the available labor ?
A)you should plant 400 acres of soy beans and no acres of corn or wheat, in this event you will use all the available labor
B) you should plant 400 acres of corn and no acres of soy or wheat, in this event you will use all the available labor
C) you should plant 400 acres of corn and no acres of soy or wheat, in this event you will not use all the available labor
D) you should plant 400 acres of soy and no acres of corn or wheat, in this event you will not use all the available labor
E) you should plant 400 acres of wheat and no acres of soy or corn, in this event you will not use all the available labor

Question

Your farm encompasses 400 acres, and you are planning to grow soy beans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: $5 for soy beans, $2 for corn, and $1 for wheat.You estimate that each acre of soy beans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of $3,000 for each acre of soy beans, $2,000 for each acre of corn, and $1,000 for each acre of wheat. You can afford to spend no more than $2,400 on fertilizer, and your farm laborers can supply 2,800 hours per week. How many acres of each crop should you plant to maximize total profits  In this event, will you be using all the available labor  ?
A)you should plant 400 acres of soy beans and no acres of corn or wheat, in this event you will use all the available labor
B) you should plant 400 acres of corn and no acres of soy or wheat, in this event you will use all the available labor
C) you should plant 400 acres of corn and no acres of soy or wheat, in this event you will not use all the available labor
D) you should plant 400 acres of soy and no acres of corn or wheat, in this event you will not use all the available labor
E) you should plant 400 acres of wheat and no acres of soy or corn, in this event you will not use all the available labor

Quizplus · Accepted Answer

To maximize total profits, we need to determine the optimal number of acres of each crop to plant. Let X, Y, and Z be the number of acres of soy, corn, and wheat to plant, respectively.

We can start by setting up the objective function, which represents the total profit:

Maximize: P = 3000X + 2000Y + 1000Z

Next, we need to set up the constraints. The first constraint is the total number of acres we can plant:

X + Y + Z = 400

The second constraint is the total cost of fertilizer we can afford:

5X + 2Y + Z ≤ 2400

The third constraint is the total amount of labor we have:

5X + 2Y + 2Z ≤ 2800

We also need to ensure that X, Y, and Z are non-negative.

Using linear programming techniques, we can solve this problem and find the optimal solution:

X = 400, Y = 0, Z = 0

This means that we should plant all 400 acres of soy and no acres of corn or wheat. In this event, we will not use all the available labor (we only need 2000 out of the 2800 available hours).

Your Farm Encompasses 400 Acres, and You Are Planning to Grow