To solve this problem, we proceed as follows:

1. Define the profit function:

The profit $P(x)$ is given by: $P(x) = R(x) - C(x)$ Substituting the given functions $R(x) = 14x$ and $C(x) = 6x + 560$: $P(x) = 14x - (6x + 560)$ Simplify: $P(x) = 14x - 6x - 560$ $P(x) = 8x - 560$

2. Solve for the required number of units:

We are asked to find $x$ when the profit is $1176. So, set $P(x) = 1176$: $8x - 560 = 1176$ Add 560 to both sides: $8x = 1736$ Divide both sides by 8: $x = 217$

Final Answer:

The business must produce 217 units to achieve a profit of $1176.

Would you like a detailed explanation of the steps or have any additional questions?

5 Related Questions:

1. How would the profit change if the cost function had an additional fixed cost?
2. Can you calculate the break-even point for this business model?
3. What happens to the profit function if the revenue per unit decreases by $2?
4. How would you graph $P(x)$, $R(x)$, and $C(x)$ on the same coordinate plane?
5. What is the profit if 300 units are produced?

Tip:

When solving profit-related problems, always confirm whether the revenue and cost functions include fixed or variable components, as they significantly impact calculations.