To find the marginal cost, we calculate the derivative of the cost function $c(q)$ with respect to $q$, denoted $c'(q)$, and then evaluate it at $q = 10$.

The cost function is given as:
$c(q) = 0.008q^3 - 2.4q^2 + 80q + 20,400$

Step 1: Differentiate $c(q)$

The derivative $c'(q)$ is: $c'(q) = \frac{d}{dq} \left( 0.008q^3 - 2.4q^2 + 80q + 20,400 \right)$ $c'(q) = 3 \cdot 0.008q^2 - 2 \cdot 2.4q + 80$ $c'(q) = 0.024q^2 - 4.8q + 80$

Step 2: Evaluate $c'(q)$ at $q = 10$

Substitute $q = 10$ into $c'(q)$: $c'(10) = 0.024(10)^2 - 4.8(10) + 80$ $c'(10) = 0.024(100) - 48 + 80$ $c'(10) = 2.4 - 48 + 80$ $c'(10) = 34.4$

Final Answer:

The marginal cost when $q = 10$ is: $\boxed{34.4}$ dollars.

Would you like a detailed breakdown of any part of the calculation?

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5 Related Questions:

1. How do you interpret the marginal cost in economics?
2. What is the significance of finding the derivative of a cost function?
3. How would the marginal cost change for a different value of $q$?
4. How can we calculate the average cost from the cost function?
5. What happens to marginal cost when the cost function is linear?

Tip:

Always check the units of your derivative; marginal cost is typically measured in cost per unit of production.