Given Total Cost and Revenue Functions

Oct 13, 2025 by magicalceylontours.com 81 views

If the Total Cost, $C = 40 + 5Q$ , and the Revenue, $R = E - 14Q - \frac{1}{2}Q^2$ , Where $Q$ is the Output, Find the Output at the Break-Even Point

Understanding the Break-Even Point

The break-even point is a crucial concept in business and economics, representing the point at which the total revenue equals the total cost. It is the point at which the business neither makes a profit nor incurs a loss. In this article, we will explore how to find the break-even point using the given total cost and revenue functions.

Given Total Cost and Revenue Functions

The total cost function is given by $C = 40 + 5Q$ , where $Q$ is the output. This function indicates that the total cost increases by $5 for every unit of output.

The revenue function is given by $R = E - 14Q - \frac{1}{2}Q^2$ , where $E$ is a constant representing the external revenue. This function indicates that the revenue decreases by $14 for every unit of output and decreases by $\frac{1}{2}$ for every additional unit of output.

Finding the Break-Even Point

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

Setting the total revenue equal to the total cost, we get:

E - 14Q - \frac{1}{2}Q^2 = 40 + 5Q

Rearranging the equation, we get:

\frac{1}{2}Q^2 + 19Q + (E - 40) = 0

This is a quadratic equation in $Q$ . We can solve this equation using the quadratic formula:

Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = \frac{1}{2}$ , $b = 19$ , and $c = E - 40$ . Plugging these values into the quadratic formula, we get:

Q = \frac{-19 \pm \sqrt{19^2 - 4(\frac{1}{2})(E - 40)}}{2(\frac{1}{2})}

Simplifying the equation, we get:

Q = \frac{-19 \pm \sqrt{361 - 2E + 80}}{1}

Q = \frac{-19 \pm \sqrt{441 - 2E}}{1}

Q = \frac{-19 \pm 21}{1}

This gives us two possible values for $Q$ :

Q_1 = \frac{-19 + 21}{1} = 2

Q_2 = \frac{-19 - 21}{1} = -40

Since $Q$ represents the output, it cannot be negative. Therefore, the only valid solution is $Q = 2$ .

Conclusion

In this article, we have explored how to find the break-even point using the given total cost and revenue functions. We have set the total revenue equal to the total cost and solved for $Q$ using the quadratic formula. The only valid solution is $Q = 2$ , which represents the output at the break-even point.

Implications of the Break-Even Point

The break-even point has significant implications for businesses. It represents the point at which the business neither makes a profit nor incurs a loss. At this point, the business is just breaking even, and any increase in output will result in a profit, while any decrease in output will result in a loss.

The break-even point is an important concept in business and economics, and it has significant implications for businesses. It represents the point at which the business is just breaking even, and any increase in output will result in a profit, while any decrease in output will result in a loss.

Real-World Applications

The break-even point has significant real-world applications. It is used by businesses to determine the optimal level of production, pricing, and investment. It is also used by investors to evaluate the potential of a business and make informed investment decisions.

In conclusion, the break-even point is an important concept in business and economics, and it has significant implications for businesses. It represents the point at which the business is just breaking even, and any increase in output will result in a profit, while any decrease in output will result in a loss.

References

Break-Even Analysis. Investopedia.
Break-Even Point. Encyclopedia Britannica.
Break-Even Analysis. AccountingTools.

Further Reading

Break-Even Analysis: A Guide for Small Business Owners. Small Business Administration.
Break-Even Point: A Guide for Entrepreneurs. Entrepreneur Magazine.
Break-Even Analysis: A Guide for Investors. Investor's Business Daily.
Break-Even Point Q&A: Frequently Asked Questions

Understanding the Break-Even Point

Q: What is the break-even point?

A: The break-even point is the point at which the total revenue equals the total cost. It is the point at which the business neither makes a profit nor incurs a loss.

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

A: To calculate the break-even point, you need to set the total revenue equal to the total cost and solve for the output (Q). The break-even point is the point at which the total revenue equals the total cost.

Q: What is the formula for calculating the break-even point?

A: The formula for calculating the break-even point is:

Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a, b, and c are the coefficients of the quadratic equation.

Q: What are the implications of the break-even point?

A: The break-even point has significant implications for businesses. It represents the point at which the business is just breaking even, and any increase in output will result in a profit, while any decrease in output will result in a loss.

Q: How do I use the break-even point in business decision-making?

A: The break-even point can be used to determine the optimal level of production, pricing, and investment. It can also be used to evaluate the potential of a business and make informed investment decisions.

Q: What are some common mistakes to avoid when calculating the break-even point?

A: Some common mistakes to avoid when calculating the break-even point include:

Not considering all costs, including fixed and variable costs
Not considering all revenue, including external revenue
Not using the correct formula for calculating the break-even point
Not considering the impact of changes in output on the break-even point

Q: How do I interpret the results of a break-even analysis?

A: To interpret the results of a break-even analysis, you need to consider the following:

The break-even point: This is the point at which the total revenue equals the total cost.
The break-even quantity: This is the quantity of output at which the break-even point occurs.
The break-even price: This is the price at which the break-even point occurs.
The sensitivity of the break-even point to changes in output: This is the impact of changes in output on the break-even point.

Q: What are some real-world applications of the break-even point?

A: Some real-world applications of the break-even point include:

Determining the optimal level of production
Pricing products and services
Evaluating the potential of a business
Making informed investment decisions

Q: How do I use the break-even point to make informed business decisions?

A: To use the break-even point to make informed business decisions, you need to consider the following:

The break-even point: This is the point at which the total revenue equals the total cost.
The break-even quantity: This is the quantity of output at which the break-even point occurs.
The break-even price: This is the price at which the break-even point occurs.
The sensitivity of the break-even point to changes in output: This is the impact of changes in output on the break-even point.

Conclusion

The break-even point is a crucial concept in business and economics, representing the point at which the total revenue equals the total cost. It is the point at which the business neither makes a profit nor incurs a loss. In this article, we have answered some frequently asked questions about the break-even point, including how to calculate it, what are the implications of the break-even point, and how to use it to make informed business decisions.

References

Break-Even Analysis. Investopedia.
Break-Even Point. Encyclopedia Britannica.
Break-Even Analysis. AccountingTools.

Further Reading

Break-Even Analysis: A Guide for Small Business Owners. Small Business Administration.
Break-Even Point: A Guide for Entrepreneurs. Entrepreneur Magazine.
Break-Even Analysis: A Guide for Investors. Investor's Business Daily.