Maximizing Bookstore Revenue
A bookstore is deciding what price it should charge for a certain book. After research, the store finds that if the book's price is $ p $ dollars (where $ p \le 32 $), then the number of books sold per month is $ 128-4p $. What price should the store charge to maximize its revenue?
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
The store's revenue is given by: number of books sold $ \times $ price of each book, or $ p(128-4p)=128p-4p^2 $. We want to maximize this expression by completing the square. We can factor out a $ -4 $ to get $ -4(p^2-32p) $.
To complete the square, we add $ (32/2)^2=256 $ inside the parentheses and subtract $ -4\cdot256=-1024 $ outside. We are left with the expression
\[-4(p^2-32p+256)+1024=-4(p-16)^2+1024.\]Note that the $ -4(p-16)^2 $ term will always be nonpositive since the perfect square is always nonnegative. Thus, the revenue is maximized when $ -4(p-16)^2 $ equals 0, which is when $ p=16 $. Thus, the store should charge $ \boxed{16} $ dollars for the book.
Alternatively, since the roots of $ p(128-4p) $ are 0 and 32, symmetry tells us that the extreme value will be at $ p=16 $. Since the coefficient on $ p^2 $ is negative, this is a maximum.
Freeflows