Final answer:
To find the marginal cost at q = 20 for the given total cost equation, differentiate the total cost function and substitute q = 20. The marginal cost at this quantity level is $90.
Step-by-step explanation:
The student's question asks for the marginal cost at a quantity (q) of 20, given the total cost (tc) equation tc = 200 + 10q + 2q2. To find this, we calculate the derivative of the total cost equation with respect to q (the quantity), which gives us the marginal cost function. Then we substitute q = 20 into this derivative to calculate the marginal cost at that quantity level.
To derive the marginal cost function, differentiate tc with respect to q to get MC = d(tc)/dq = 10 + 4q. Now, substitute q = 20 into the MC equation to find the marginal cost at q = 20: MC = 10 + 4(20) = 10 + 80 = 90. The marginal cost at q = 20 is therefore $90.