Problem

1: Chapters 1 to 4
The total revenue was $\$$

Answer

Expert–verified
Hide Steps
Answer

\(\boxed{20}\) dollars should be charged for the book to maximize the revenue.

Steps

Step 1 :First, we need to find the expression for the total revenue. The total revenue is given by the number of books sold times the price of each book.

Step 2 :Let the price of the book be \(p\) dollars, and the number of books sold be \(120-3p\). Then the total revenue is \(p(120-3p)\).

Step 3 :Now, we need to maximize the expression \(120p-3p^2\).

Step 4 :Factor out a \(-3\) to get \(-3(p^2-40p)\).

Step 5 :To complete the square, add \((40/2)^2=400\) inside the parenthesis and subtract \(-3\cdot400=-1200\) outside. We get \(-3(p^2-40p+400)+1200\).

Step 6 :Simplify the expression to \(-3(p-20)^2+1200\).

Step 7 :The \(-3(p-20)^2\) term will always be nonpositive since the perfect square is always nonnegative. Thus, the revenue is maximized when \(-3(p-20)^2\) equals 0.

Step 8 :This occurs when \(p=20\).

Step 9 :\(\boxed{20}\) dollars should be charged for the book to maximize the revenue.

link_gpt