Question

The total costs (in thousands of dollars) for producing certain commodities can be modeled by C(q)=a(q¹/³+1)²+b,(a,b are constants ) where q is the number of commodities produced in hundreds. The fixed cost is 20,000 dollars, and the total cost when 100 commodities are produced is 23,000 dollars. 1. Find the values of a and b. 2. Find the marginal cost function C′(q). Evaluate C′(1) and interpret its meaning in this context. 3. The average cost of production is given by Cˉ(q)=qC(q)​ What happens to the average cost Cˉ(q) if the number of commodities produced goes to infinity?

Answer 1

1. The values of a and b are a = 10 and b = 20.

2. The marginal cost function C'(q) is given by
`\(C'(q) = (2a)/(3)q^(-2/3)\)`. Evaluating C'(1), we get
`\(C'(1) = (2a)/(3)\)`, and in this context, it represents the instantaneous rate of change of total cost with respect to the number of commodities produced.

3. As the number of commodities produced goes to infinity, the average cost Cˉ(q) approaches the marginal cost. Therefore, Cˉ(q) approaches
`\((2a)/(3)q^(-2/3)\).`

Step-by-step explanation:

1. To find the values of a and b, we use the given information that the fixed cost is $20,000, and the total cost when 100 commodities are produced is $23,000. Substituting these values into the cost function, we obtain the equations
`\(20,000 = a(100^(1/3) + 1)^2 + b\)` and
`23,000 = a(100^(1/3) + 1)^2 + b\)`. Solving these equations simultaneously, we find a = 10 and b = 20.

2. The marginal cost function C'(q) is obtained by finding the derivative of the total cost function with respect to q. Evaluating C'(1) gives us the specific value of the marginal cost at q = 1, which represents the rate of change of cost when one additional commodity is produced.

3. The average cost of production Cˉ(q) is the total cost divided by the number of commodities produced. As q approaches infinity, the average cost tends to the marginal cost, indicating that in the long run, the cost per commodity stabilizes, and producing additional commodities has a diminishing impact on average cost. This convergence reflects economies of scale, suggesting that as production scales up, the average cost per unit decreases.