Respuesta :
Answer:
The cost of producing 100 items is $723.35
Step-by-step explanation:
The marginal cost is the derivative of the total cost function, so we have
[tex]C^{'}(x)=1.65-0.002x[/tex]
To find the total cost function we need to do integration
[tex]C(x)= \int\, C^{'}(x)dx \\C(x)=\int\,(1.65-0.002x) dx[/tex]
Apply the sum rule to find the integral
[tex]\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)[/tex]
[tex]\int \:1.65dx=1.65x\\\int \:0.002xdx=0.001x^2[/tex]
[tex]C(x)=\int\,(1.65-0.002x) dx = 1.65x-0.001x^2+D [/tex]
D is the constant of integration
We are given that C(1) = $570, we can use this to find the value of the constant in the total cost function
[tex]C(1)=570=1.65*(1)+0.001*(1)^2+D\\D=570-1.649=568.351[/tex]
So the total cost function is [tex]C(x)=1.65x-0.001x^2+568.351 [/tex] and the cost of producing 100 items is
x=100
[tex]C(100)=1.65*(100)-0.001*(100)^2+568.351 = 723.35[/tex]
