The short-run total cost function, for a firm producing skateboards, is TC=20q22-10q+80, where q is the number of skateboards per week

Given the total cost function:

TC = 20q22 -10q + 80

Fixed cost = $80

Variablecost=2q2−10qVariablecost=2q2−10q

Marginalcost=∂TC∂q=40q−10Marginalcost=∂TC∂q=40q−10

Average total cost

TCq=Averagetotalcost=20q2−10q+80qTCq=Averagetotalcost=20q2−10q+80q

ATC=20q−10+80qATC=20q−10+80q

To find the minimum ATC, we have to differentiate the average total cost function with respect to 'q',

∂ATC∂q=20+80(−1q2)=0∂ATC∂q=20+80(−1q2)=0

80q2=2080q2=20

80=20q280=20q2

q2=4q2=4

q=2q=2

At q*=2, the average total cost is minimum.

To find the minimum average total cost, put q=2 in the ATC function calculated above,

ATC=20×2−10+802ATC=20×2−10+802

=40−10+40=$70