Question

Marginal Cost Suppose that the dollar cost of producing \(x\) washing machines is \(c(x)=2000+100 x-0.1 x^{2} .\) (a) Find the average cost of producing 100 washing machines. (b) Find the marginal cost when 100 machines are produced. (c) Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Short answer

The average cost of producing 100 washing machines is $27. The marginal cost when 100 machines are produced is $80. This is indeed approximately equal to the cost of producing one more washing machine after 100 have been produced, which is $80.10.

Step-by-step solution

Calculate Average Cost

To calculate the average cost of producing 100 washing machines, we use the formula: \nAverage Cost = Total Cost / Quantity \nBy substituting the given values we have Average Cost = c(100) / 100 = (2000+100*100-0.1*100^2) / 100 = $27$.

Calculate Marginal Cost

The marginal cost function is found by taking the derivative of the cost function. The cost function c(x) is given as 2000+100x-0.1x^2. The derivative, c'(x), gives us the marginal cost function. Thus, the marginal cost function, c'(x), is 100 - 0.2x. Therefore, the marginal cost when 100 machines are produced will be c'(100) = 100 - 0.2*100 = $80$.

Verify Marginal Cost

To confirm that the marginal cost at 100 washing machines is the cost of one more machine after the first 100, compute the difference between cost functions of 101 and 100 machines. This should approximately equal the marginal cost at 100 machines. Specifically, c(101)-c(100) = (2000 + 100*101 - 0.1*101^2) - (2000 +100*100 - 0.1*100^2) = $80.10$ which is approximately the marginal cost we calculated in step 2.