022 Exam 1 Sample A, Problem 9

Find the marginal cost to produce one more item if the fixed cost is $400, the variable cost formula is $x^{2}+30x$, and the current production quantity is 9 units.

Foundations:
Recall that the total cost, $C(x)$, is the sum of the fixed and variable costs. Also, the marginal cost at $x_{0}$ units is defined to be the effective cost of the next unit produced, and is precisely $C'(x_{0})$.

Solution:
Note that
$C(x)\,\,=\,\,{\mbox{fixed costs + variable costs}}\,\,=\,\,400+x^{2}+30x.$
The expression of the marginal cost is then
$C'(x)\,\,=\,\,2x+30.$
Finally, at a production quantity of 9 units, we have the marginal cost
$C'(9)\,\,=\,\,2(9)+30\,\,=\,\,48.$
Hence, the marginal cost at a production quantity of 9 units is $48.

Final Answer:
The marginal cost at a production quantity of 9 units is $48.

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