The marginal cost function of a product is given by MC = 2(2x + 9)^{-1/2} where x is the number of units produced. Find the average cost of production for 8 units of production, if... The marginal cost function of a product is given by MC = 2(2x + 9)^{-1/2} where x is the number of units produced. Find the average cost of production for 8 units of production, if the fixed cost is Rs 4.
Understand the Problem
The question is asking to find the average cost of production given a marginal cost function and fixed costs. We need to integrate the marginal cost function to get the total cost, then divide by the number of units produced to find the average cost.
Answer
The average cost of production for 8 units is Rs $34$.
Answer for screen readers
The average cost of production for 8 units is Rs 34.
Steps to Solve
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Define the Marginal Cost Function The marginal cost function is given by: $$ MC = 2(2x + 9) - \frac{1}{2} $$
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Integrate the Marginal Cost Function To find the total cost function, we need to integrate the marginal cost function. The integral of $MC$ will give us the total cost ($C$) as a function of $x$: $$ C(x) = \int (2(2x + 9) - \frac{1}{2}) , dx $$
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Compute the Integral Carrying out the integration step by step:
- Split the function: $$ C(x) = \int (4x + 18 - \frac{1}{2}) , dx $$
- This simplifies to: $$ C(x) = \int (4x + 17.5) , dx $$ Now integrate: $$ C(x) = 2x^2 + 17.5x + C_0 $$
- Determine the Constant of Integration To find the constant $C_0$, we need to evaluate the total cost function at $x = 0$. Since we have fixed costs of Rs 4: $$ C(0) = C_0 = 4 $$
So, our total cost function is: $$ C(x) = 2x^2 + 17.5x + 4 $$
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Calculate Total Cost for 8 Units Substitute $x = 8$ into the total cost function: $$ C(8) = 2(8^2) + 17.5(8) + 4 $$
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Compute the Cost Calculate:
- $2(64) + 17.5(8) + 4 = 128 + 140 + 4 = 272$
- Find the Average Cost Finally, to find the average cost, divide the total cost by the number of units produced: $$ \text{Average Cost} = \frac{C(8)}{8} = \frac{272}{8} = 34 $$
The average cost of production for 8 units is Rs 34.
More Information
The average cost provides insight into the production efficiency of the product. This analysis combines fixed costs with marginal costs to give a more accurate financial outlook.
Tips
- Forgetting to include fixed costs in the total cost calculation.
- Miscalculating the integration step or the final substitution.
- Not properly factoring in the number of units when calculating average cost.
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