1. **State the problem:** We want to find the number of TV sets $x$ to produce and sell per week to maximize the weekly profit, given the cost function and selling price. 2. **Define profit function:** Profit $P(x)$ is revenue minus cost. Revenue = price per set $\times$ number of sets = $85x$. Cost $C(x) = 1500 + 10x + 0.005x^2$. So, $$P(x) = 85x - (1500 + 10x + 0.005x^2) = 85x - 1500 - 10x - 0.005x^2$$ Simplify: $$P(x) = (85x - 10x) - 1500 - 0.005x^2 = 75x - 1500 - 0.005x^2$$ 3. **Maximize profit:** Since $P(x)$ is a quadratic function with a negative coefficient for $x^2$, it opens downward and has a maximum at its vertex. The vertex $x$-value is given by: $$x = -\frac{b}{2a}$$ where $P(x) = ax^2 + bx + c$, here $a = -0.005$, $b = 75$. Calculate: $$x = -\frac{75}{2 \times (-0.005)} = -\frac{75}{-0.01} = 7500$$ 4. **Check constraints:** Maximum production is 10,000 sets, and $7500 \leq 10000$, so $x=7500$ is feasible. 5. **Conclusion:** The manufacturer should produce and sell **7500** TV sets per week to maximize profit. **Final answer:** 7500