A Factory Has Fixed Costs Of $\$1,275$ Per Month. The Cost Of Producing Each Unit Of Its Product Is $\$2.50$. Each Unit Sells For $\$10$.The Equations That Model The Cost And Revenue For Producing $x$ Units Are

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Understanding the Problem

A factory has fixed costs of $1,275\$1,275 per month, which are expenses that remain the same regardless of the number of units produced. The cost of producing each unit of its product is $2.50\$2.50, and each unit sells for $10\$10. In this article, we will analyze the cost and revenue equations for producing xx units and determine the break-even point, where the cost equals the revenue.

Cost Equation

The cost equation represents the total cost of producing xx units. It includes both the fixed costs and the variable costs, which are the costs that vary with the number of units produced. The fixed costs are $1,275\$1,275 per month, and the variable costs are $2.50\$2.50 per unit. Therefore, the cost equation can be represented as:

C(x) = 1275 + 2.5x

where C(x) is the total cost of producing xx units.

Revenue Equation

The revenue equation represents the total revenue generated from selling xx units. Since each unit sells for $10\$10, the revenue equation can be represented as:

R(x) = 10x

where R(x) is the total revenue generated from selling xx units.

Break-Even Point

The break-even point is the point at which the cost equals the revenue. To find the break-even point, we need to set the cost equation equal to the revenue equation and solve for xx.

1275 + 2.5x = 10x

Subtracting 2.5x from both sides gives:

1275 = 7.5x

Dividing both sides by 7.5 gives:

x = 170

Therefore, the break-even point is 170 units.

Profit Equation

The profit equation represents the total profit generated from selling xx units. It is the difference between the revenue and the cost. Therefore, the profit equation can be represented as:

P(x) = R(x) - C(x) = 10x - (1275 + 2.5x) = 7.5x - 1275

where P(x) is the total profit generated from selling xx units.

Marginal Cost and Marginal Revenue

The marginal cost is the change in the total cost when one more unit is produced. It is the cost of producing one more unit. The marginal cost can be represented as:

MC = dC/dx = 2.5

The marginal revenue is the change in the total revenue when one more unit is sold. It is the revenue generated from selling one more unit. The marginal revenue can be represented as:

MR = dR/dx = 10

Optimizing Production

To optimize production, we need to find the level of production that maximizes the profit. This can be done by setting the marginal revenue equal to the marginal cost and solving for xx.

10 = 2.5

Since the marginal revenue is greater than the marginal cost, it is profitable to produce more units. However, as the number of units produced increases, the marginal cost also increases. Therefore, we need to find the optimal level of production that balances the marginal revenue and the marginal cost.

Solving for Optimal Production

To solve for the optimal production level, we need to find the point at which the marginal revenue equals the marginal cost. This can be done by setting the marginal revenue equal to the marginal cost and solving for xx.

10 = 2.5 + (1275 / x)

Simplifying the equation gives:

7.5 = 1275 / x

Multiplying both sides by x gives:

7.5x = 1275

Dividing both sides by 7.5 gives:

x = 170

Therefore, the optimal production level is 170 units.

Conclusion

In this article, we analyzed the cost and revenue equations for producing xx units and determined the break-even point, where the cost equals the revenue. We also found the profit equation and the marginal cost and marginal revenue. Finally, we optimized production by finding the level of production that maximizes the profit. The optimal production level is 170 units.

References

Understanding the Problem

In our previous article, we analyzed the cost and revenue equations for producing xx units and determined the break-even point, where the cost equals the revenue. We also found the profit equation and the marginal cost and marginal revenue. In this article, we will answer some frequently asked questions related to the cost and revenue analysis.

Q: What is the break-even point?

A: The break-even point is the point at which the cost equals the revenue. It is the point at which the factory is neither making a profit nor incurring a loss.

Q: How do I calculate the break-even point?

A: To calculate the break-even point, you need to set the cost equation equal to the revenue equation and solve for xx. The cost equation is C(x) = 1275 + 2.5x, and the revenue equation is R(x) = 10x. Setting the two equations equal to each other gives:

1275 + 2.5x = 10x

Solving for xx gives:

x = 170

Q: What is the profit equation?

A: The profit equation represents the total profit generated from selling xx units. It is the difference between the revenue and the cost. The profit equation can be represented as:

P(x) = R(x) - C(x) = 10x - (1275 + 2.5x) = 7.5x - 1275

Q: How do I calculate the marginal cost and marginal revenue?

A: The marginal cost is the change in the total cost when one more unit is produced. It is the cost of producing one more unit. The marginal cost can be represented as:

MC = dC/dx = 2.5

The marginal revenue is the change in the total revenue when one more unit is sold. It is the revenue generated from selling one more unit. The marginal revenue can be represented as:

MR = dR/dx = 10

Q: How do I optimize production?

A: To optimize production, you need to find the level of production that maximizes the profit. This can be done by setting the marginal revenue equal to the marginal cost and solving for xx.

10 = 2.5 + (1275 / x)

Simplifying the equation gives:

7.5 = 1275 / x

Multiplying both sides by x gives:

7.5x = 1275

Dividing both sides by 7.5 gives:

x = 170

Q: What is the optimal production level?

A: The optimal production level is the level of production that maximizes the profit. In this case, the optimal production level is 170 units.

Q: How do I determine the optimal price?

A: The optimal price is the price at which the revenue equals the cost. To determine the optimal price, you need to set the revenue equation equal to the cost equation and solve for the price.

10x = 1275 + 2.5x

Solving for the price gives:

x = 170

The optimal price is therefore:

P = 10x / 170 = 0.588

Q: What is the optimal price in dollars?

A: The optimal price in dollars is:

P = 0.588 * 170 = 100

Conclusion

In this article, we answered some frequently asked questions related to the cost and revenue analysis. We calculated the break-even point, the profit equation, the marginal cost and marginal revenue, and the optimal production level. We also determined the optimal price in dollars.

References