Question

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 $.

Answer

Given

P(x)=190-0.5xC(x)=2500+0.75x2

The total Revenue is

R(x)=P(x)x=190-0.5xx=190x-0.5x2

The total​ profit is

P=R-C=190x-0.5x2-2500+0.75x2=190x-0.5x2-2500-0.75x2=-1.25x2+190x-2500

To find number of suits must the company produce and sell in order to maximize​ profit differentiate P=-1.25x2+190x-2500 with respect to x we get

P'=-1.25(2x)+190-0=-2.50x+190

For finding maximum number of suits 

P'=0 -2.50x+190=0 2.50x=190 x=1902.50 x=76

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

Maximum profit is

P(76)=-1.25(76)2+190(76)-2500=-1.25(5776)+14,440-2500=-7220+11,940=4720

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

p=190-0.5(76)=190-38=152

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