1. **State the problem:** We want to find the least number of copies sold so that the publishing company makes a positive profit. 2. **Define variables and given data:** Let $x$ be the number of copies sold. Cost per copy = 0.38 Revenue from dealers per copy = 0.35 Advertising revenue = 10% of dealer revenue for copies sold beyond 10,000 units. 3. **Write the profit formula:** Profit = Total Revenue - Total Cost Total Cost = $0.38x$ Total Revenue = Revenue from dealers + Advertising revenue Revenue from dealers = $0.35x$ Advertising revenue = $0.10 \times 0.35 \times (x - 10000)$ for $x > 10000$, else 0 So, $$\text{Profit} = 0.35x + 0.10 \times 0.35 \times (x - 10000) - 0.38x \quad \text{for } x > 10000$$ For $x \leq 10000$, advertising revenue is 0, so $$\text{Profit} = 0.35x - 0.38x = -0.03x \leq 0$$ which is never positive. 4. **Simplify the profit expression for $x > 10000$:** $$\text{Profit} = 0.35x + 0.035(x - 10000) - 0.38x$$ $$= 0.35x + 0.035x - 350 - 0.38x$$ $$= (0.35 + 0.035 - 0.38)x - 350$$ $$= 0.005x - 350$$ 5. **Set profit > 0 to find minimum $x$: ** $$0.005x - 350 > 0$$ $$0.005x > 350$$ $$x > \frac{350}{0.005}$$ $$x > 70000$$ 6. **Conclusion:** The least number of copies that must be sold to have a positive profit is $70001$ (since $x$ must be an integer and greater than 70000).