Question

​Raggs, Ltd. a clothing​ firm, determines that in order to sell x​ suits, the price per suit must be pequals160 minus 0.75 x. It also determines that the total cost of producing x suits is given by Upper C (x )equals4000 plus 0.5 x squared. ​a) Find the total​ revenue, Upper R (x ). ​b) Find the total​ profit, Upper 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?

Answer 1

Answer:

Please refer to the below for answers.

Explanation:

As per the above information, the following can be deduced;

•The price per suit (P)= 160-0.75x

•Total cost of producing x suit is given as C(x) = 4,000+0.5x^2

(a) The total revenue is R(x);

= (Numbers of units) * (Price per unit)

R(x) = x * (160-0.75x)

R(x) = 160x - 0.75x^2

(b) The total profit is P(x);

= Total revenue - Total costs

P(x) = R(x) - C(x)

P(x) = (160x - 0.75x^2) - (4,000 + 0.5x^2)

P(x) = 160x - 0.75x^2 - 4,000 - 0.5x^2

P(x) = -1.25x^2 + 160x - 4,000

(c). To find the maximum value of P(x), we will need to find first the derivative of P'(x)

d/dx*P(x) = d/dx(-1.25x^2 + 160x - 4,000)

P'(x) = - d/dx(1.25x^2) + d/dx(160x) - d/DX(4,000)

P'(x) = -2.5x + 160

The next is to find the critical points.

P'(x) = 0

-2.5x + 160 = 0

-2.5x . 10 + 160 . 10 = 0 . 10

-25x + 1,600 = 0

-25x + 1,600 - 1,600 = 0 - 1,600

-25x = -1,600

x = 64

We will also use the second derivative test to know if there is an absolute maximum because f(c) is the absolute maximum value if f''(c) < 0

Therefore,

d/dx*P'(x) = d/dx (-2.5x + 160)

P''(x) = -d/dx (-2.5x) + d/dx (160)

P''(x) = -2.5

Hence, (64) is negative , and so profit is maximized when 64 suits are produced and sold.

(d) The maximum profit is denoted as;

P(64) = -1.25(64)^2 + 160(64) - 4,000

P(64) = -64^2 * 1.25 + 10,240 - 4,000

P(64) = 6,240 - 64^2 * 1.25

P(64) = 6,240 - 5,120

P(64) = $1,120

Therefore , the clothing firms makes $1,120 as profit by producing and selling 64 suits.

(e) The price per unit to needed to make the maximum profit is thus;

P = 160 - 0.75x

Where x is 64

P = 160 - 0.75(64)

P = 160 - 48

P = $112