Question

Cost, revenue, and profit are in dollars and x is the number of units. If the marginal cost for producing a product is

MC=86−2e −⁰.⁰¹ˣ with a fixed cost of $6,200, find the total cost function. C(x)=__

Answer 1

The total cost function is found by integrating the marginal cost function and adding the fixed cost, resulting in C(x) = 86x - 200e^{-0.01x} + 6200.

Step-by-step explanation:

To find the total cost function for producing a product, we begin with the given marginal cost (MC) function, MC = 86 - 2e-0.01x, and a fixed cost of $6,200. Since the marginal cost represents the cost to produce one more unit of a product, we integrate the marginal cost function with respect to x to find the total variable cost. Then, we add the fixed cost to find the total cost function.

1. Integrate the marginal cost function: ∫ (86 - 2e-0.01x) dx.
2. Add the fixed cost to the integrated function to determine the total cost function C(x).

The integration step yields: C(x) = 86x - 200e-0.01x + C, where C is the constant of integration. Since we know the fixed cost is $6,200, this constant is the fixed cost, making the final total cost function: C(x) = 86x - 200e-0.01x + 6200.