A cereal factory has weekly fixed costs of $48,000. It costs $1.37 to produce each box of cereal. A box of cereal sells for $3.99. Express the profit function p(x) that gives the total weekly profit from x boxes of cereal.
cost for x boxes = 1.37x
fixed cost regardless of x: 48000
Income for x boxes = 3.99x
Profit = income - total costs
p(x) = 3.99x - (48000 + 1.37x)
Oh, the cereal business, where your hopes are high and your flakes are higher! Now, let me crunch those numbers for you. To find the profit function, we need to consider the costs and the revenue.
Cost function: It costs $1.37 to produce each box, and if we make x boxes, the total cost would be c(x) = 1.37x.
Revenue function: A box of cereal sells for $3.99, so the revenue for x boxes would be r(x) = 3.99x.
Now, the profit function p(x) represents the difference between revenue and cost, so we have:
p(x) = r(x) - c(x)
= 3.99x - 1.37x
= 2.62x
So, the profit function p(x) that gives the total weekly profit from x boxes of cereal is 2.62x. Just remember, it's a-MAIZE-ing to see those profits roll in! Keep up the good work!
To find the profit function, we need to subtract the total cost from the total revenue.
Total cost = Fixed costs + Cost per box * Number of boxes produced
Total cost = $48,000 + $1.37x
Total revenue = Price per box * Number of boxes sold
Total revenue = $3.99x
Profit function = Total revenue - Total cost
Profit function = $3.99x - ($48,000 + $1.37x)
Simplifying the equation, we get:
Profit function p(x) = $3.99x - $48,000 - $1.37x
To express the profit function, we need to calculate the total cost and the total revenue.
The total cost is calculated by multiplying the cost per box of cereal ($1.37) by the number of boxes produced (x). Since the costs are fixed, we also need to consider the weekly fixed cost of $48,000. Therefore, the total cost is given by the formula:
Total Cost = (Cost per box * Number of boxes) + Weekly Fixed Cost
= ($1.37x) + $48,000
Next, we calculate the total revenue by multiplying the selling price per box of cereal ($3.99) by the number of boxes sold (x). The revenue is given by the formula:
Total Revenue = Selling Price per box * Number of boxes
= $3.99x
Finally, we can express the profit function, denoted as p(x), which gives the total weekly profit from x boxes of cereal. The profit is calculated by subtracting the total cost from the total revenue:
Profit = Total Revenue - Total Cost
= $3.99x - ($1.37x + $48,000)
= $3.99x - $1.37x - $48,000
Simplifying the equation, we get:
p(x) = $2.62x - $48,000
Therefore, the profit function is p(x) = $2.62x - $48,000.