Question

We are promoting a new book. The price was originally set at $26 and an average of 825 books were sold monthly. After research, it was found that for every $1 decrease in price, 75 more books would be sold. What is the price that will maximize sales? What is the maximum projected sales? What price would yield no sales

Answer 1

Answer:

Step-by-step explanation:

The price that will maximize sales and the number of projected sales can be found in the vertex of a quadratic equation.  We need to use the puzzle of the info given to somehow create a set of biomials that can be multiplied together to get this quadratic.  We will separate the number of books sold from the price of the books in a table of sorts:

                # of books                              $ per book

We know that a number of 825 books was sold when the price per book was $26.

                # of books                             $ per book

                      825                                       $26

If we have a $1 decrease in price per book, let's use as our unknown the number of $1 decreases in price per book.  In other words, x = # of $1 decreases.  

If we decrease the price per book by $1, we are decreasing the price by 1x which is 1 dollar.

If, when we decrease the price per book by $1, we are selling the original 825 books plus another 75 books at the $1 decrease per book.  Now we have in our table

              # of books                             $ per book

               825 + 75x                                  26 - x

Under # of books, the expression in words says, "we are still selling 825 books, but now we are adding an additional 75 books per month when we lower the cost $1 per book".

Under $ per book, the expression in words says, "the cost per book is the starting cost minus $1".

To get the quadratic that results from this, multiply the 2 expressions together to get

$P(x)=-75x^2+1125x+21450$ or, if we want to keep things positive:

$P(x)=75x^2-1125x-21450$

We can factor a 75 out of each term to make the numbers a bit smaller:

$P(x)=75(x^2-15x-286)$

Use the quick formulas for h and k to solve for the vertex.  You could complete the square to get there, but once you know these formulas, there's no need to go through the very long long process of completing the square.

$h=-\frac{b}{2a}$ and $k=c-\frac{b^2}{4a}$

These formulas come from the quadratic equation.  h here will give us the number of $1 decreases we need to maximize k, the profit from this amount decreases.

Our a = 1, b = -15 and c = -286.  Filling in for h:

$h=-\frac{-15}{2}$ and

$h=\frac{15}{2}$ so

h = 7.5  That is the number of decreases

Filling in for k:

$k=-286-\frac{(-15)^2}{4}$ and

$k=-286-\frac{225}{4}$ so

k = 342.25  That is the maximum projected sales from this amount of decreases in price.  

But we are not done.  We need to use the 7.5 to find out what the best price would be to maximize the sales and therefore, the profit.  Going back to our cost expression, 26 - x, using 7.5 for x:

26 - 7.5 = $18.50

This means that if we sell books at $18.50 rather than $26, we can expect to earn $342.25 per month in book sales.