A sportswear manufacture determines that the marginal cost of producing x worm-up suits is given in dollars by 20-0.015x.if the cost of producing one suits is $25,find the cost function and the cost of producing 50 suits and 100 suits
To find the cost function, we need to derive it from the given marginal cost function. The marginal cost is the rate of change of the cost with respect to the number of suits produced.
Given: Marginal cost (MC) = 20 - 0.015x
Now, let's integrate the marginal cost function to find the total cost function (C).
C = ∫(MC) dx
C = ∫(20 - 0.015x) dx
C = 20x - 0.015*(x^2)/2 + C1
Here, C1 is the constant of integration. Since we are given that the cost of producing one suit is $25, we can substitute this information to find the value of C1.
When x = 1, C = 25
25 = 20(1) - 0.015(1^2)/2 + C1
25 = 20 - 0.0075/2 + C1
C1 = 25 - 20 + 0.0075/2
C1 = 4.00375
Therefore, the total cost function is:
C = 20x - 0.0075x^2 + 4.00375
To find the cost of producing 50 suits, substitute x = 50 into the total cost function:
C(50) = 20(50) - 0.0075(50^2) + 4.00375
C(50) = 1000 - 18.75 + 4.00375
C(50) = 985.25375
The cost of producing 50 suits is $985.25.
Similarly, to find the cost of producing 100 suits, substitute x = 100 into the total cost function:
C(100) = 20(100) - 0.0075(100^2) + 4.00375
C(100) = 2000 - 75 + 4.00375
C(100) = 1929.00375
The cost of producing 100 suits is $1929.00.