A manufacture of CD player has monthly fixed cost of 8600 and variable cost of 55 per unit for one particular model. The company sells this model to dealers for 115 each. Which is their function for the profit. Please show all work.
income = 115 n
cost = 8600 + 55 n
so
profit = 115 n - 8600 - 55 n
= 60 n - 8600
That means we do not start making money until we sell 8600/60 = 143 1/3 of the players. After that we will make 60 dollars per unit.
To find the profit function, we need to understand the components that contribute to profit. Profit is calculated by subtracting the total cost from the total revenue.
Total Cost:
The total cost consists of fixed costs and variable costs.
Fixed Costs: The fixed costs for this CD player model are given as $8600 per month. Since fixed costs remain constant regardless of the number of units produced or sold, the fixed cost can be considered as a constant value.
Variable Costs: The variable cost per unit is given as $55. This means that for every unit produced, the cost of producing it is $55.
Total Revenue:
The total revenue is obtained by multiplying the selling price per unit by the number of units sold.
Selling Price per Unit: The selling price to dealers for each unit is given as $115.
Now we can use this information to find the profit function. Let's define 'x' as the number of units produced/sold.
Total Cost Function:
The total cost function (TC) can be calculated by adding the fixed costs and the variable costs.
TC = Fixed Costs + (Variable Cost per Unit * Number of Units)
TC = 8600 + (55 * x)
Total Revenue Function:
The total revenue function (TR) can be calculated by multiplying the selling price per unit by the number of units sold.
TR = Selling Price per Unit * Number of Units
TR = 115x
Profit Function:
The profit function (P) can be calculated by subtracting the total cost from the total revenue.
P = TR - TC
P = (115x) - (8600 + 55x)
Simplifying the equation further:
P = 115x - 8600 - 55x
P = 60x - 8600
Therefore, the profit function for the CD player manufacturer is P(x) = 60x - 8600, where x represents the number of units produced/sold.