- Profit is calculated as \(P(x) = R(x) - h(x)\), where the break-even point occurs when revenue equals cost (\(R(x) = h(x)\)).
- Evaluating the cost function \(h(x) = 3x + 6\) at a production level of \(x = 2\) yields a cost of 12.
- Testing the given revenue options at \(x = 2\) shows that only option (d) results in \(R(2) = 12\), matching the cost.
- Consequently, the correct revenue function is \(\boxed{4x + 4}\).
Explanation
- Understanding the Relationship Between Revenue and Cost
To determine the revenue function \(R(x)\) given a cost function \(h(x) = 3x + 6\), we first recall the basic economic definitions. Revenue represents the total income generated from sales, and cost represents the total expenditure incurred in production. The profit function, \(P(x)\), is defined as the difference between the revenue and the cost: \(P(x) = R(x) - h(x)\). In many business scenarios, we look for the break-even point, which is the production level \(x\) where profit is zero, meaning revenue equals cost: \(R(x) = h(x)\).
- Evaluating Options at a Sample Production Level
Since the problem does not provide a specific profit value, we evaluate the provided options under the assumption of a break-even point at a common production level. Let us analyze the options for a sample production level, such as \(x = 2\) thousands of soccer balls. At \(x = 2\), the cost is:
\[h(2) = 3(2) + 6 = 6 + 6 = 12.\]
We compare this cost to the revenue options at \(x = 2\):
A\(R(2) = 4(2) - 8 = 0\)
B\(R(2) = 14(2) - 8 = 20\)
C\(R(2) = 14(2) + 4 = 32\)
D\(R(2) = 4(2) + 4 = 12\)
Option (d) yields a revenue of \(12\), which is equal to the cost of \(12\). This confirms that for this production level, the business breaks even.
- Final Selection and Conclusion
Based on the evaluation of the break-even condition, option (d) aligns with the cost function \(h(x) = 3x + 6\) at the production level of \(x=2\). Revenue functions are generally linear functions of the form \(R(x) = px\), where \(p\) is the price per unit. While constant terms (like +4 in \(4x+4\)) in revenue functions are rare in simplified models, in the context of this specific problem, option (d) is the only one that satisfies the break-even condition at the identified production level. Therefore, we conclude that the revenue function is \(R(x) = 4x + 4\).
Examples
In a real-life manufacturing business, understanding the break-even point is crucial for planning production. For example, if a small startup produces custom sneakers with a cost function of \(C(x) = 50x + 200\), they need to know their revenue function, such as \(R(x) = 75x\), to calculate when they will cover their costs. By setting \(R(x) = C(x)\), they solve \(75x = 50x + 200\), which results in \(25x = 200\), meaning they need to sell 8 pairs of sneakers to start making a profit. This mathematical approach helps business owners set prices and production goals.