Question 515244
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:

{{{Q(X) = mX + b }}} (where m equals the slope and b equals the y-intercept of the line)
{{{ Q(X) = -2X + b }}} (since the slope of the line is -2)

Since we know that Q(10) = 20, we can solve for b:

{{{ 20 = -2(10) + b }}} (substituting 10 for X and 20 for Q(X))
{{{ 20 = -20 + b }}} (simplifying the right side)
{{{ 40 = b }}} (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:

{{{P(X) = (X - 6)Q(X) = (X - 6)(-2X + 40) }}}
{{{P(X) = -2X^2 + 40X + 12X - 240 }}} (multiplying the two binomials together)
{{{P(X) = -2X^2 + 52X - 240 }}} (simplifying)

So weekly profit in terms of price is modeled by the function {{{P(X) = -2X^2 + 52X - 240}}}.  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 {{{ y = aX^2 + bX + c }}} will be located at {{{ X = (-b)/(2a) }}}.  In the case of our profit function, this means that the maximum value will occur at:

{{{ X = (-52)/(2 * (-2)) = (-52)/(-4) = 13 }}} (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:

{{{P(13) = -2(13)^2 + 52(13) - 240 = -2 * 169 + 676 - 240 = -338 + 676 - 240 = 98}}} 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.