Question 515244: A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $6, and the society sells an average of 20 per week at a price of $10 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week.
a) Find a function that models weekly profit in terms of price per feeder.
b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?
Answer by drcole(72)   (Show Source):
You can put this solution on YOUR website! Let X be the price of the feeder. First, let's find a function Q(X) giving the average quantity of feeders sold per week at price X. We know that Q(10) = 20 (that is, at a price of $10, the society sells 20 bird feeders per week on average). We also know that for every $1 increase in price, the society loses two sales per week. This is a linear function: if we graphed this function, it would look like a line with slope -2 (each dollar increase causes the quantity sold to decrease by 2). So Q(X) has the form:
 (where m equals the slope and b equals the y-intercept of the line)
 (since the slope of the line is -2)
Since we know that Q(10) = 20, we can solve for b:
 (substituting 10 for X and 20 for Q(X))
 (simplifying the right side)
 (adding 20 to both sides)
So the formula for the average quantity of bird feeders sold at price X is Q(X) = -2X + 40.
Next, we compute P(X), the weekly profit. Each feeder costs $6 to make, so at a price of X dollars, the profit from each bird feeder would be X - 6 dollars. The weekly profit is going to be the profit per bird feeder times the weekly quantity of feeders sold, or Q(X), so we have that:
 (multiplying the two binomials together)
 (simplifying)
So weekly profit in terms of price is modeled by the function  . This is a quadratic function, and its graph looks like a parabola that is open downward. We can use calculus to find the maximum value, or we can simply remember that the vertex of the parabola  will be located at  . In the case of our profit function, this means that the maximum value will occur at:
 (where we used b = 52 and a = -2)
So the society should charge $13 per bird feeder to maximize profits. At that price, our model tells us that the weekly profit will be:
 dollars
Does this make sense? At $10, the society was averaging 20 bird feeders sold per week, with a profit of $4 per feeder, giving a weekly profit of $80. At $11, they would make a profit of $5 per feeder, but only sell 18 feeders on average, giving a weekly profit of $90. At $12, the profit per feeder would rise to $6, but the number sold per week on average would fall to 16, yielding a weekly profit of $96. At $13, profits would now be $7 per feeder, but only 14 would be sold on average per week, making the average weekly profit $98. If we increase the price again to $14, the society earns a profit of $8 per feeder, but would sell only 12 feeders on average per week, pushing the weekly profit back down to $96. So $13 does appear to be the price that maximizes profit.
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