Final answer:
The linear cost function, based on a fixed cost of $200 and a variable cost found by deducting this fixed cost from the total cost for 30 items and dividing by 30, is C(x) = $200 + $60x, where x is the number of items produced.
Step-by-step explanation:
The student is asked to find a linear cost function based on a given fixed cost and the cost of producing a certain number of items. The fixed cost is $200 and the cost to produce 30 items is $2,000. To determine the cost function, we first need to calculate the variable cost per item. This is done by subtracting the fixed cost from the total cost for 30 items, and then dividing by the number of items:
Total variable cost = Total cost to produce 30 items - Fixed cost = $2,000 - $200 = $1,800.
Variable cost per item = Total variable cost / Number of items = $1,800 / 30 = $60.
Now, we can express the cost function as:
Cost function (C) = Fixed cost + (Variable cost per item × Number of items).
Symbolically, this is:
C(x) = $200 + $60x,
where x is the number of items produced and C(x) is the total cost of production.
This function shows that at zero production, the fixed costs of $200 are still present. With an increase in production, the variable cost is added to the fixed cost. Therefore, the total cost is the sum of fixed and variable costs, with the fixed cost being the vertical intercept of the total cost curve.