Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be p = 180 − 0.5x. It also determines that the total cost of producing x suits is given by C(x) = 4500 + 0.75 x².
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) Revenue obtained from selling x suits

R(x) = 180x - 0.5x²

b) Total Profit obtained from selling x suits

P(x) = -1.25x² + 180x - 4500

c) The number of units that should be produced and sold to maximize profits is 72 units.

d) Maximum Profit = $1,980

e) Price per suit that must be charged in order to maximize profit = $144

Step-by-step explanation:

Assume the price is in dollars.

Price per suit is given as

p = 180 − 0.5x

Cost of producing x suits

C(x) = 4500 + 0.75 x²

a) Revenue obtained from selling x suits

R(x) = px = x (180 − 0.5x) = 180x - 0.5x²

b) Total Profit = Revenue - Cost

= (180x - 0.5x²) - (4500 + 0.75x²)

= 180x - 1.25x² - 4500

P(x) = -1.25x² + 180x - 4500

c) in order to maximize profit, we find the maximum of the profit function.

At the maximum point, (dP/dx) = 0

P(x) = -1.25x² + 180x - 4500

(dP/dx) = -2.5x + 180 = 0

x = (180/2.5) = 72 units.

Therefore, the number of units that should be produced and sold to maximize profits is 72 units.

d) The maximum profit

To obtain this, we just slot in the number of units that should be produced and sold to maximize profits into the Profits function.

P(x) = -1.25x² + 180x - 4500

x = 72 units

P(x) = -1.25(72²) + 180(72) - 4500 = $1980

e) Price per suit that must be charged in order to maximize profit

Price per suit is given as

p = 180 − 0.5x

Slotting the number of units that should be produced and sold to maximize profits into this expression,

p = 180 - 0.5(72) = $144

Hope this Helps!!!

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