A clothing firm determines that in order to sell x suits, the price per suit must be p = 150 - 0.5x. It also determines that the total cost of producing x suits is given by C(x) = 4000 + 0.25x^2.
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 make this maximum profit?
To find the total revenue R(x), we need to multiply the number of suits (x) by the price per suit (p). The price per suit is given by p = 150 - 0.5x.
a. Total revenue R(x) = x * p = x * (150 - 0.5x)
Simplifying, R(x) = 150x - 0.5x^2.
To find the total profit P(x), we need to subtract the total cost of producing x suits (C(x)) from the total revenue (R(x)).
b. Total profit P(x) = R(x) - C(x)
Substituting the expressions we found, P(x) = (150x - 0.5x^2) - (4000 + 0.25x^2)
Simplifying, P(x) = 150x - 0.5x^2 - 4000 - 0.25x^2
Combining like terms, P(x) = -0.75x^2 + 150x - 4000.
To find the number of suits that maximize profit, we need to find the x-value (number of suits) that corresponds to the vertex of the quadratic equation P(x) = -0.75x^2 + 150x - 4000. The x-value of the vertex is given by -b/2a, where a = -0.75 and b = 150.
c. Number of suits to maximize profit = -b/2a = -150/(2 * -0.75)
Simplifying, Number of suits to maximize profit = 100.
To find the maximum profit, we substitute the value of x (number of suits) that maximizes profit into the profit equation P(x) = -0.75x^2 + 150x - 4000.
d. Maximum profit = P(100) = -0.75(100)^2 + 150(100) - 4000
Simplifying, Maximum profit = -7500 + 15000 - 4000
Maximum profit = $3500.
To find the price per suit that must be charged to make this maximum profit, we substitute the value of x (number of suits) that maximizes profit into the price equation p = 150 - 0.5x.
e. Price per suit for maximum profit = p(100) = 150 - 0.5(100)
Simplifying, Price per suit for maximum profit = 150 - 50
Price per suit for maximum profit = $100.
Okay...
x=the number of suits sold
price=150-0.5x
cost=C(x)=4000+0.25x^2
a) Revenue is the total money made which is just the price times the number of suits sold, hence:
R(x)=(150-0.5x)(x)
b) Profit is the amount of money made or revenue minus the cost, hence:
P(x)=R(x)-C(x)