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# Raggs, Ltd. a clothing​ firm, determines that in order to sell x​ suits, the price per suit must be p=190−0.5x. It...

Raggs, Ltd. a clothing​ firm, determines that in order to sell x​ suits, the price per suit must be
p=190−0.5x.
It also determines that the total cost of producing x suits is given by
C(x)=2500+0.75x2.
​a) Find the total​ revenue,
R(x).
​b) Find the total​ profit,
P(x).
​c) How many suits must the company produce and sell in order to maximize​ profit?
​d) What is the maximum​ profit?
​e) What price per suit must be charged in order to maximize​ profit?
Transcribed: Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be p = 190 – 0.5x. It also determines that the total cost of producing x suits is given by C(x) = 2500 + 0.75x. a) Find the total revenue, R(x). b) Find the total profit, P(x). c) How many suits must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per suit must be charged in order to maximize profit? a) R(x) = b) P(x) = c) suits d) The maximum profit is \$ e) The price per unit must be \$.

Given

$P\left(x\right)=190-0.5x\phantom{\rule{0ex}{0ex}}C\left(x\right)=2500+0.75{x}^{2}$

The total Revenue is

$\begin{array}{rcl}R\left(x\right)& =& P\left(x\right)x\\ & =& \left[190-0.5x\right]x\\ & =& 190x-0.5{x}^{2}\end{array}$

The total​ profit is

$\begin{array}{rcl}P& =& R-C\\ & =& 190x-0.5{x}^{2}-\left[2500+0.75{x}^{2}\right]\\ & =& 190x-0.5{x}^{2}-2500-0.75{x}^{2}\\ & =& -1.25{x}^{2}+190x-2500\end{array}$

To find number of suits must the company produce and sell in order to maximize​ profit differentiate $P=-1.25{x}^{2}+190x-2500$ with respect to x we get

$\begin{array}{rcl}P\text{'}& =& -1.25\left(2x\right)+190-0\\ & =& -2.50x+190\end{array}$

For finding maximum number of suits

So, 76 suits must the company produce and sell in order to maximize​ profit

Maximum profit is

$\begin{array}{rcl}P\left(76\right)& =& -1.25{\left(76\right)}^{2}+190\left(76\right)-2500\\ & =& -1.25\left(5776\right)+14,440-2500\\ & =& -7220+11,940\\ & =& 4720\end{array}$

To find price per suit must be charged in order to maximize​ profit

$\begin{array}{rcl}p& =& 190-0.5\left(76\right)\\ & =& 190-38\\ & =& 152\end{array}$

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