# The manager of a CD store has found that if the price of a CD is p(x)= 75-x/6 then x CDs will be sold. An expression for the total revenue from the sale of x CDs is R(x) =75x-x^2/6 How do you find the number of CDs that will produce maximum revenue?

Mar 10, 2018

$225$ CDs will produce the maximum revenue.

#### Explanation:

We know from Calculus that, for ${R}_{\max}$, we must have,

$R ' \left(x\right) = 0 , \mathmr{and} , R ' ' \left(x\right) < 0$.

Now, $R \left(x\right) = 75 x - {x}^{2} / 6 \Rightarrow R ' \left(x\right) = 75 - \frac{1}{6} \cdot 2 x = 75 - \frac{x}{3}$.

$\therefore R ' \left(x\right) = 0 \Rightarrow \frac{x}{3} = 75 , \mathmr{and} , x = 75 \cdot 3 = 225$.

Further, $R ' \left(x\right) = 75 - \frac{x}{3} \Rightarrow R ' ' \left(x\right) = - \frac{1}{3} < 0 , \text{ already}$.

Hence, $x = 225 \text{ gives } {R}_{\max}$.

Thus, $225$ CDs will produce the maximum revenue ${R}_{\max}$.

color(magenta)(BONUS :

${R}_{\max} = R \left(225\right) = 75 \cdot 225 - {225}^{2} / 6 = 8437.5 , \mathmr{and}$

$\text{Price of a CD=} p \left(225\right) = 75 - \frac{225}{6} = 37.5$.