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
