1. **State the problem:** We have two drones at points C and D on the ground. The drone at C follows the path $f(x) = -x^2 + 2x + q$, and the drone at D follows $g(x) = mx + 5$. We know $CD = 7$ meters and $AB = 2$ meters. Point E is directly above C, and we need to find the length $EC$. 2. **Analyze given information:** - Point C lies on the x-axis, so its y-coordinate is 0. - Since $f(x)$ models the flight path of the drone at C, and it passes through C on the x-axis, $f(x_C) = 0$. - Point A is on the y-axis above C, and $AB = 2$ meters, so $A$ is at $(0, q)$ and $B$ is at $(0, 2)$. 3. **Find $q$ using point A:** Since $A$ is on the parabola $f(x)$ and on the y-axis, its x-coordinate is 0. Calculate $f(0)$: $$f(0) = -0^2 + 2\cdot0 + q = q$$ Given $A$ is at $(0, q)$ and $AB = 2$, and $B$ is at $(0, 2)$, so: $$|q - 2| = 2$$ This gives two possibilities: - $q - 2 = 2 \Rightarrow q = 4$ - $2 - q = 2 \Rightarrow q = 0$ 4. **Find $x$-coordinate of C:** Since $C$ lies on the x-axis, $f(x_C) = 0$: $$-x_C^2 + 2x_C + q = 0$$ Try $q=4$: $$-x_C^2 + 2x_C + 4 = 0 \Rightarrow x_C^2 - 2x_C - 4 = 0$$ Discriminant: $$\Delta = (-2)^2 - 4 \cdot 1 \cdot (-4) = 4 + 16 = 20$$ Roots: $$x_C = \frac{2 \pm \sqrt{20}}{2} = 1 \pm \sqrt{5}$$ Since $C$ is on the x-axis and $D$ is at $(7,0)$ with $CD=7$, the distance between $C$ and $D$ is: $$|x_D - x_C| = 7$$ Given $x_D = 7$, check for $x_C$: - If $x_C = 1 + \sqrt{5} \approx 3.236$, then $7 - 3.236 = 3.764 \neq 7$ - If $x_C = 1 - \sqrt{5} \approx -1.236$, then $7 - (-1.236) = 8.236 \neq 7$ Try $q=0$: $$-x_C^2 + 2x_C = 0 \Rightarrow x_C(-x_C + 2) = 0$$ Roots: $$x_C = 0 \text{ or } x_C = 2$$ Check distances: - If $x_C=0$, then $CD = 7 - 0 = 7$ correct. - If $x_C=2$, then $CD = 7 - 2 = 5$ incorrect. So $q=0$ and $x_C=0$. 5. **Find length $EC$:** Point E is directly above C, so $E$ has coordinates $(0, f(0)) = (0, 0)$ since $q=0$. But $f(0) = 0$, so $E$ is at $(0,0)$, same as $C$. This contradicts the problem statement that $E$ is above $C$. 6. **Re-examine $AB=2$:** Given $B$ is at $(0,2)$ and $AB=2$, so $A$ is at $(0,4)$ or $(0,0)$. Since $A$ is on the parabola $f(x)$, and $f(0) = q$, then $q=4$ or $q=0$. 7. **Use $q=4$ and $x_C = 1 - \sqrt{5} \approx -1.236$:** Calculate $CD$: $$|7 - (-1.236)| = 8.236 \neq 7$$ Calculate $CD$ for $x_C = 1 + \sqrt{5} \approx 3.236$: $$|7 - 3.236| = 3.764 \neq 7$$ 8. **Try to find $m$ from $g(x) = mx + 5$ passing through $B(0,2)$ and $D(7,0)$:** At $x=0$, $g(0) = 5$, but $B$ is at $(0,2)$, so $g(0) = 5 \neq 2$. This is inconsistent. 9. **Adjust $g(x)$ to pass through $B(0,2)$ and $D(7,0)$:** Find slope $m$: $$m = \frac{0 - 2}{7 - 0} = \frac{-2}{7}$$ Equation: $$g(x) = -\frac{2}{7}x + 2$$ 10. **Conclusion:** Given inconsistencies, the best consistent values are: - $q=0$ - $x_C=0$ - $E$ is at $(0, f(0)) = (0,0)$ - Length $EC = |f(0) - 0| = 0$ Since $E$ is directly above $C$ and $f(0) = 0$, the length $EC = 0$ meters. **Final answer:** $$EC = 0 \text{ meters}$$