Question

Assume that the situation can be expressed as a linear cost function. Find the cost function. Marginal cost: $ 30 ; 170 items cost $ 7500 to produce.

Answer 1

The cost function, given the marginal cost of $30 and that 170 items cost $7500 to produce, is found by using the linear cost function formula and solving for the y-intercept. The resulting cost function is C(x) = 30x + 2400.

Step-by-step explanation:

To find the cost function when the marginal cost is $30 and the cost to produce 170 items is $7500, we will need to follow these steps:

1. Understand the Definition of Marginal Cost: Marginal cost is the additional cost to produce one more unit of a product. For linear cost functions, marginal cost is constant and equals the slope of the cost function.
2. Set up the Linear Cost Function: A linear cost function can be written in the form C(x) = mx + b, where C(x) is the total cost to produce x items, m is the slope (marginal cost), and b is the y-intercept (fixed costs).
3. Use Given Information to Solve for 'b': With the marginal cost being $30 (m = 30), and knowing that producing 170 items costs $7500 (C(170) = 7500), we can substitute these values into our linear equation to solve for 'b'.

Calculating Fixed Costs:

• Substitute the known values into the cost function: 7500 = 30(170) + b.
• Solve for b: 7500 = 5100 + b,
• Find b: b = 7500 - 5100 = $2400.

Thus, the cost function is C(x) = 30x + 2400.