A farm supply store carries 50-lb bags of both grain pellets and grain mash for pig feed. It can store 600 bags of pig feed. At least twice as many of its customers prefer the mash to the pellets. The store buys the pellets for $3.75 per bag and sells them for $6.00. It buys the mash for $2.50 per bag and sells it for $4.00. If the store orders no more that $1400 worth of pig feed, how many bags of mash should the store order to make the most profit?
Answer: 400 bags
To solve this problem, we need to determine the number of bags of mash that the store should order to maximize its profit.
Let's assume that the number of bags of mash the store orders is x. Since the store carries 600 bags of pig feed in total, the number of bags of pellets can be calculated as (600 - x).
According to the problem statement, at least twice as many customers prefer the mash to the pellets. This means that the number of bags of pellets should be less than or equal to half of the number of bags of mash. Mathematically, this can be written as:
(600 - x) ≤ (1/2) * x
Simplifying the inequality, we get:
1200 - 2x ≤ x
Next, let's calculate the cost and selling price of the bags of mash and pellets. The cost of the pellets is $3.75 per bag, and the selling price is $6.00 per bag. Therefore, the profit per bag of pellets is:
Profit per bag of pellets = Selling price - Cost price = $6.00 - $3.75 = $2.25
The cost of the mash is $2.50 per bag, and the selling price is $4.00 per bag. Therefore, the profit per bag of mash is:
Profit per bag of mash = Selling price - Cost price = $4.00 - $2.50 = $1.50
Now, let's calculate the total cost and the total selling price of the pig feed. The total cost can be determined by multiplying the cost per bag by the number of bags, while the total selling price can be obtained by multiplying the selling price per bag by the number of bags:
Total cost = (600 - x) * $3.75 + x * $2.50
Total selling price = (600 - x) * $6.00 + x * $4.00
According to the problem statement, the total cost should be less than or equal to $1400. So, we can set up the following inequality:
(600 - x) * $3.75 + x * $2.50 ≤ $1400
Now, we need to find the value of x that satisfies both inequalities. To determine the maximum profit, we can create a profit function, which is the total selling price minus the total cost:
Profit = Total selling price - Total cost
Substituting the expressions for the total selling price and total cost, the profit function becomes:
Profit = [(600 - x) * $6.00 + x * $4.00] - [(600 - x) * $3.75 + x * $2.50]
Simplifying the expression, we get:
Profit = $1.50x + $1200
To find the value of x that maximizes the profit, we take the derivative of the profit function with respect to x and set it equal to 0:
d(Profit)/dx = $1.50 - 0 = 0
Solving for x, we get:
$1.50x = -$1200
x = 400
Therefore, the store should order 400 bags of mash to make the most profit.