Question

At a price of p $perunit,themarketingdepartmentinacompanyestimatesthattheweeklycost$C$andrevenue$R$\left(inthousandsofdollars\right)willbegivenbytheequations$C = 28 - 2p$Costequation$R = 9p - {p}^{2}$ Revenue equation
Find the prices for which the company has
(A) A profit (B) A loss

Answer 1

Given :

$\text{Cost function}\left[C\right(p\left)\right]=28-2p$

$\text{Revenue function}\left[R\right(p\left)\right]=9p-p^2$

For profit:

$R\left(p\right)>C\left(p\right)$

$\Rightarrow 9p-p^2>28-2p$

$\Rightarrow 9p-p^2-28+2p>0$

$\Rightarrow p^2-11p+28<0$

$\Rightarrow p\in ($4, $7)$

For loss:

$C\left(p\right)>R\left(p\right)$

$\Rightarrow 28-2p>9p-p^2$

$\Rightarrow p^2-11p+28>0$

$\Rightarrow p\in (-\infty ,$4)\cup ($7,\infty )$