1. **Problem Statement:** Two planes, a Cessna and a Boeing 747, are flying toward the same intersection point at the same altitude. The Cessna is 100 miles north of the intersection moving south at 120 mph. The 747 is 550 miles east of the intersection moving west at 600 mph. 2. **Parametric Equations for the Planes:** Let $t$ be time in hours, with $t=0$ at the moment described. - Cessna position: moving south from 100 miles north, so $$x_C(t) = 0, \quad y_C(t) = 100 - 120t$$ - 747 position: moving west from 550 miles east, so $$x_B(t) = 550 - 600t, \quad y_B(t) = 0$$ 3. **Distance Between the Planes as a Function of Time:** Distance $D(t)$ is given by the distance formula: $$D(t) = \sqrt{(x_C(t) - x_B(t))^2 + (y_C(t) - y_B(t))^2}$$ Substitute: $$D(t) = \sqrt{(0 - (550 - 600t))^2 + ((100 - 120t) - 0)^2} = \sqrt{(600t - 550)^2 + (100 - 120t)^2}$$ 4. **Graphing the Distance Function:** The function to graph is: $$D(t) = \sqrt{(600t - 550)^2 + (100 - 120t)^2}$$ for $t \geq 0$. 5. **Minimum Distance and When It Occurs:** To find the minimum distance, minimize $D(t)^2$ to avoid the square root: $$D(t)^2 = (600t - 550)^2 + (100 - 120t)^2$$ Expand: $$= (600t)^2 - 2 \cdot 600t \cdot 550 + 550^2 + 10000 - 2 \cdot 100 \cdot 120t + (120t)^2$$ $$= 360000t^2 - 660000t + 302500 + 10000 - 24000t + 14400t^2$$ Combine like terms: $$= (360000 + 14400) t^2 - (660000 + 24000) t + (302500 + 10000)$$ $$= 374400 t^2 - 684000 t + 312500$$ Take derivative and set to zero: $$\frac{d}{dt} D(t)^2 = 2 \cdot 374400 t - 684000 = 0$$ $$748800 t = 684000$$ $$t = \frac{684000}{748800} = \frac{95}{104} \approx 0.9135 \text{ hours}$$ Calculate minimum distance: $$D(0.9135) = \sqrt{(600 \cdot 0.9135 - 550)^2 + (100 - 120 \cdot 0.9135)^2}$$ $$= \sqrt{(548.1 - 550)^2 + (100 - 109.62)^2} = \sqrt{(-1.9)^2 + (-9.62)^2}$$ $$= \sqrt{3.61 + 92.54} = \sqrt{96.15} \approx 9.81 \text{ miles}$$ 6. **Simultaneous Graphing of the Planes' Paths:** - Cessna path: vertical line $x=0$, $y=100 - 120t$ - 747 path: horizontal line $y=0$, $x=550 - 600t$ **Final answers:** - (a) $x_C(t) = 0$, $y_C(t) = 100 - 120t$; $x_B(t) = 550 - 600t$, $y_B(t) = 0$ - (b) $D(t) = \sqrt{(600t - 550)^2 + (100 - 120t)^2}$ - (c) Graph $D(t)$ for $t \geq 0$ - (d) Minimum distance is approximately 9.81 miles at $t \approx 0.9135$ hours - (e) Graph parametric equations simultaneously to simulate motion