Question

At a price of p $perunit,themarketingdepartmentinacompanyestimatesthattheweeklycost$C$andrevenue$R$\left(inthousandsofdollars\right)willbegivenbytheequations$C = 27 - 2p$Costequation$R = 10p - {p}^{2}$ Revenue equation
Find the prices for which the company has
(A) A profit (B) A loss

Answer 1

Given :

$\text{Cost function}\left[C\right(p\left)\right]=27-2p$

$\text{Revenue function}\left[R\right(p\left)\right]=10p-p^2$

For profit:

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

$\Rightarrow 10p-p^2>27-2p$

$\Rightarrow 10p-p^2-27+2p>0$

$\Rightarrow p^2-12p+27<0$

$\Rightarrow p\in ($3, $9)$

For loss:

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

$\Rightarrow 27-2p>10p-p^2$

$\Rightarrow p^2-12p+27>0$

$\Rightarrow p\in (-\infty ,$3)\cup ($9,\infty )$