To find the break-even quantity, we need to determine the value of $x$ where the cost $C(x)$ equals the revenue $R(x)$. In other words, we solve the equation:

$C(x) = R(x)$

Given: $C(x) = 13x + 11800$ $R(x) = 213x$

Setting these equal: $13x + 11800 = 213x$

Now, let's isolate $x$:

1. Subtract $13x$ from both sides: $11800 = 200x$

2. Divide both sides by 200 to solve for $x$: $x = \frac{11800}{200} = 59$

Therefore, the break-even quantity is: $x = 59$

Answer:

The break-even quantity is $x = 59$.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What is the total cost at the break-even point?
2. What is the revenue at the break-even point?
3. How would the break-even quantity change if the fixed cost increased?
4. What happens to the break-even quantity if the revenue per unit decreases?
5. Can there be multiple break-even points for non-linear cost and revenue functions?

Tip: The break-even point is found where total cost equals total revenue, which can reveal the minimum sales needed to avoid a loss.