Question

Mr. Avinash manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is ` 30. On the other hand, the cost, in rupees, of producing x units is $240 + bx + cx^2$ , where b and c are some constants. Avinash noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66.66 percent. However, an increase in daily production from 40 to 60 units results in an increase of only 50 percent in the daily production cost. Assume that demand is unlimited and that Avinash can sell as much as he can produce. His objective is to maximize the profit. How many units should Mr. Avinash produce daily?

• A. 70
• B. 100
• C. 130
• D. 150
• E. 180

Answer 1

The Correct Option is B

To solve this problem, we need to determine the optimal number of units Mr. Avinash should produce to maximize his profit. Let's analyze the given cost function and other conditions step by step. 

Step 1: Define the Cost Function

The cost of producing \(x\) units is given by:

\(C(x) = 240 + bx + cx^2\)

Step 2: Use Given Conditions to Find Constants b and c

1. Doubling production from 20 to 40 units increases the production cost by 66.66%.

2. Increasing production from 40 to 60 units increases production cost by 50%.

The cost for 20 units:

\(C(20) = 240 + 20b + 400c\)

The cost for 40 units:

\(C(40) = 240 + 40b + 1600c\)

The increase in cost from 20 to 40 units is:

\(C(40) - C(20) = (240 + 40b + 1600c) - (240 + 20b + 400c) = 20b + 1200c\)

Given this increase is 66.66%, we have:

\(\frac{20b + 1200c}{240 + 20b + 400c} = \frac{66.66}{100}\)

The cost for 60 units:

\(C(60) = 240 + 60b + 3600c\)

The increase in cost from 40 to 60 units is:

\(C(60) - C(40) = (240 + 60b + 3600c) - (240 + 40b + 1600c) = 20b + 2000c\)

Given this increase is 50%, we have:

\(\frac{20b + 2000c}{240 + 40b + 1600c} = \frac{50}{100}\)

Solving these two equations allows us to find the values of \(b\) and \(c\).

Step 3: Profit Maximization

Profit is maximized when the marginal cost equals marginal revenue (price per unit):

Marginal Cost = Derivative of \(C(x)\).

Marginal Revenue = Selling Price = 30.

Set the derivative of the cost \((b + 2cx)\) equal to 30 and solve for \(x\).

Step 4: Calculate and Verify

Substitute the values of \(b\) and \(c\) found earlier into the derivative:

\(b + 2cx = 30\)

Solve for \(x\) to find the production quantity that maximizes profit.

Step 5: Conclusion

After calculating, the optimal number of units Mr. Avinash should produce daily is 100.

This analysis leads us to the answer.