The problem involves a fast-food restaurant's production costs and revenue. We're given:

• The cost function $C(x) = 6x + 120$, where $x$ represents the number of units produced.
• The revenue function $R(x) = 13x$.

The company starts to turn a profit when its revenue exceeds its cost. This occurs when:

$R(x) > C(x)$

Substituting the given functions:

$13x > 6x + 120$

Now, let's solve for $x$:

$13x - 6x > 120$ $7x > 120$ $x > \frac{120}{7}$ $x > 17.14$

Since the company starts turning a profit after producing a whole number of units, we round $x$ up to the nearest whole number, which is 18.

Thus, the company starts to turn a profit when the production level reaches at least 18 units.

Would you like more details, or have any questions about this solution?

Relative Questions:

1. How do fixed costs affect a company's break-even point?
2. What would happen if the cost function had a different slope?
3. How is profit maximization different from the break-even analysis?
4. How do changes in revenue affect the production threshold for profit?
5. What if the cost function were non-linear—how would the calculation change?

Tip:

Always check if the profit function involves rounding or additional conditions, as small details can affect the final solution.