Question

The cost (in dollars) of producing x units of a certain commodity is

C(x)=100+50x-0.02x².
(a) Find the average rate of change of C with respect to x when the production level is
changed from x = 100 to 105.

Answer 1

The average rate of change of the cost function C(x) when production increases from 100 units to 105 units is 45.90 dollars per unit, which is the marginal cost over that range.

Step-by-step explanation:

The student is asking about the average rate of change of the cost function C(x) = 100 + 50x - 0.02x² when production increases from x = 100 units to 105 units. To find this, we calculate the change in total cost and divide by the change in quantity.

First, let's compute the cost for x = 100 and x = 105:

C(100) = 100 + 50(100) - 0.02(100)² = 100 + 5000 - 200 = 4900

C(105) = 100 + 50(105) - 0.02(105)² = 100 + 5250 - 220.50 = 5129.50

The difference in cost (ΔC) is 5129.50 - 4900 = 229.50 dollars, and the difference in quantity (Δx) is 105 - 100 = 5 units.

The average rate of change (ΔC/Δx) is:

229.50 / 5 = 45.90 dollars per unit.

This result represents the average additional cost of producing one more unit of output over the interval from 100 to 105 units, which can be considered as the marginal cost over that range.