1. **State the problem:** We want to find the maximum value of $p$ such that the probability of no faulty chips $P(\text{no fault})$ is at least 0.92, given the approximate expression from part b: $$P(\text{no fault}) \approx a + bp + cp^2$$ where $n=200$ and $p < 0.001$. 2. **Recall the approximate expression from part b:** From the binomial expansion, $$P(\text{no fault}) = (1-p)^n \approx 1 - np + \frac{n(n-1)}{2}p^2$$ Substituting $n=200$, $$P(\text{no fault}) \approx 1 - 200p + \frac{200 \times 199}{2} p^2 = 1 - 200p + 19900 p^2$$ So, $$a=1, \quad b=-200, \quad c=19900$$ 3. **Set up the inequality for the desired probability:** We want $$P(\text{no fault}) \geq 0.92$$ Using the approximation, $$1 - 200p + 19900 p^2 \geq 0.92$$ 4. **Simplify the inequality:** $$1 - 200p + 19900 p^2 - 0.92 \geq 0$$ $$0.08 - 200p + 19900 p^2 \geq 0$$ 5. **Rewrite as a quadratic inequality:** $$19900 p^2 - 200 p + 0.08 \geq 0$$ 6. **Solve the quadratic equation:** Set the quadratic equal to zero to find critical points: $$19900 p^2 - 200 p + 0.08 = 0$$ Use the quadratic formula: $$p = \frac{200 \pm \sqrt{(-200)^2 - 4 \times 19900 \times 0.08}}{2 \times 19900}$$ Calculate the discriminant: $$\Delta = 40000 - 4 \times 19900 \times 0.08 = 40000 - 6368 = 33632$$ $$\sqrt{33632} \approx 183.38$$ 7. **Calculate the roots:** $$p_1 = \frac{200 - 183.38}{39800} = \frac{16.62}{39800} \approx 0.000417$$ $$p_2 = \frac{200 + 183.38}{39800} = \frac{383.38}{39800} \approx 0.00963$$ 8. **Analyze the inequality:** Since the quadratic coefficient $19900 > 0$, the parabola opens upwards. The inequality $\geq 0$ holds outside the roots, i.e., for $$p \leq 0.000417 \quad \text{or} \quad p \geq 0.00963$$ 9. **Apply the domain restriction:** Given $p < 0.001$, only the interval $$p \leq 0.000417$$ is valid. 10. **Final answer:** The maximum value of $p$ to achieve at least 92% probability of no faulty chips is approximately $$\boxed{0.00042}$$ This means the probability of a single chip being faulty must be less than or equal to about 0.00042 for the company to meet its target.