document.write( "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.\r
\n" ); document.write( "\n" ); document.write( "a) Find a function that models weekly profit in terms of price per feeder.\r
\n" ); document.write( "\n" ); document.write( "b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit? \n" ); document.write( "
Algebra.Com's Answer #343876 by drcole(72)\"\" \"About 
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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:\r
\n" ); document.write( "\n" ); document.write( "\"Q%28X%29+=+mX+%2B+b+\" (where m equals the slope and b equals the y-intercept of the line)
\n" ); document.write( "\"+Q%28X%29+=+-2X+%2B+b+\" (since the slope of the line is -2)\r
\n" ); document.write( "\n" ); document.write( "Since we know that Q(10) = 20, we can solve for b:\r
\n" ); document.write( "\n" ); document.write( "\"+20+=+-2%2810%29+%2B+b+\" (substituting 10 for X and 20 for Q(X))
\n" ); document.write( "\"+20+=+-20+%2B+b+\" (simplifying the right side)
\n" ); document.write( "\"+40+=+b+\" (adding 20 to both sides)\r
\n" ); document.write( "\n" ); document.write( "So the formula for the average quantity of bird feeders sold at price X is Q(X) = -2X + 40.\r
\n" ); document.write( "\n" ); document.write( "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:\r
\n" ); document.write( "\n" ); document.write( "\"P%28X%29+=+%28X+-+6%29Q%28X%29+=+%28X+-+6%29%28-2X+%2B+40%29+\"
\n" ); document.write( "\"P%28X%29+=+-2X%5E2+%2B+40X+%2B+12X+-+240+\" (multiplying the two binomials together)
\n" ); document.write( "\"P%28X%29+=+-2X%5E2+%2B+52X+-+240+\" (simplifying)\r
\n" ); document.write( "\n" ); document.write( "So weekly profit in terms of price is modeled by the function \"P%28X%29+=+-2X%5E2+%2B+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%5E2+%2B+bX+%2B+c+\" will be located at \"+X+=+%28-b%29%2F%282a%29+\". In the case of our profit function, this means that the maximum value will occur at:\r
\n" ); document.write( "\n" ); document.write( "\"+X+=+%28-52%29%2F%282+%2A+%28-2%29%29+=+%28-52%29%2F%28-4%29+=+13+\" (where we used b = 52 and a = -2)\r
\n" ); document.write( "\n" ); document.write( "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:\r
\n" ); document.write( "\n" ); document.write( " dollars\r
\n" ); document.write( "\n" ); document.write( "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. \n" ); document.write( "
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