A company has been selling 1200 computer games per week at $18 each. Data indicates that for each $1 price increase, there will be a loss of 40 sales per week. It costs $10 to produce each game.
a) State the revenue function
b) State the cost function
c) State the profit function
d) What price will produce the maximum profit
To answer these questions, we need to understand and define the functions that will help us calculate the revenue, cost, and profit.
a) Revenue Function:
The revenue is calculated by multiplying the number of units sold by the price per unit. In this case, the number of units sold per week is given as 1200, and the price per unit is $18. The revenue function can be written as follows:
Revenue = (Number of Units Sold) * (Price per Unit)
Revenue = 1200 * 18
Revenue = 21,600
Therefore, the revenue function is R(x) = 21,600, where x represents the number of units sold.
b) Cost Function:
The cost per unit for producing the games is given as $10. Since the cost is constant per unit, the cost function can be written as follows:
Cost = (Cost per Unit) * (Number of Units Sold)
Cost = 10 * x
Therefore, the cost function is C(x) = 10x, where x represents the number of units sold.
c) Profit Function:
Profit is calculated by subtracting the cost from the revenue. So, the profit function can be written as:
Profit = Revenue - Cost
Profit = R(x) - C(x)
Profit = 21,600 - 10x
Therefore, the profit function is P(x) = 21,600 - 10x, where x represents the number of units sold.
d) Maximum Profit:
To find the price that will produce the maximum profit, we need to analyze the profit function. In this case, the profit function is a linear function where the coefficient of x represents the decrease in profit for each unit sold. Since the price increase of $1 leads to a loss of 40 sales per week, it means that the profit decreases by $40 for each unit sold.
Therefore, we can represent the coefficient of x as -40.
To find the price that will produce the maximum profit, we can use the formula:
x = -b / (2a)
In this case, a is -10 (the coefficient of x in the profit function) and b is 0 (since there is no constant term).
x = -0 / (2(-10))
x = 0 / (-20)
x = 0
Therefore, the price that will produce the maximum profit is $0, meaning that the company should not charge anything for the computer games in order to maximize profit. However, this result might not be practical, so further analysis and considerations would be required.