The materials for each feeder cost $6, and they sell an average of 20 per week at a price of $10 each. They conduct a survey and find that for every dollar increase they lose 2 sales per week. Find a function that models weekly profit in terms of price per feeder
To find a function that models the weekly profit in terms of price per feeder, we need to break down the different components involved.
Let's denote the price per feeder as "x."
First, let's calculate the number of feeders sold based on the price per feeder. We know that for every dollar increase, two sales are lost. This means that for every dollar decrease, two sales are gained.
If we initially sell 20 feeders at $10 each, we can express the number of feeders sold as a function of the price per feeder:
Number of feeders sold = 20 - 2 * (x - 10)
Next, we need to calculate the total revenue generated from the sales. The revenue is calculated by multiplying the number of feeders sold by the price per feeder:
Revenue = (20 - 2 * (x - 10)) * x
The cost of materials for each feeder is given as $6. Therefore, the total cost of materials for the feeders sold would be:
Cost of materials = 6 * (20 - 2 * (x - 10))
Finally, to obtain the weekly profit, we subtract the total cost of materials from the total revenue:
Profit = (20 - 2 * (x - 10)) * x - 6 * (20 - 2 * (x - 10))
The function that models the weekly profit in terms of the price per feeder is:
Profit(x) = (20 - 2 * (x - 10)) * x - 6 * (20 - 2 * (x - 10))
Simplifying further, we get:
Profit(x) = (20 - 2x + 20) * x - (6 * 20 - 12 * (x - 10))
Profit(x) = 40x - 2x^2 - 120 + 12x - 120 + 12x
Profit(x) = -2x^2 + 64x - 240
So, the function that models the weekly profit in terms of the price per feeder is Profit(x) = -2x^2 + 64x - 240.