Question

1. two planes take off from the same airstrip. the first plane flies west for 150 miles and then flies 30° south of west for 220 miles. the second plane flies east for 220 miles and then flies ( x^circ ) south of east for 150 miles. if ( x < 30 ), which plane is farther from the airstrip after the second leg? justify your answer.

Explanation:

Step1: Define displacement vectors for Plane 1

Let the airstrip be at the origin. West is $-x$ direction, South is $-y$ direction.
First leg: $\vec{v_1} = \langle -150, 0
angle$
Second leg: $220$ miles at $30^\circ$ south of west:
$\vec{v_2} = \langle -220\cos30^\circ, -220\sin30^\circ
angle$
Total displacement $\vec{D_1} = \vec{v_1}+\vec{v_2} = \langle -150 - 220\cos30^\circ, 0 - 220\sin30^\circ
angle$

Step2: Calculate magnitude for Plane 1

Use distance formula $|\vec{D}| = \sqrt{D_x^2 + D_y^2}$:

$$\begin{align*} |\vec{D_1}| &= \sqrt{(-150 - 220\cos30^\circ)^2 + (-220\sin30^\circ)^2}\\ &= \sqrt{150^2 + 2\cdot150\cdot220\cos30^\circ + (220\cos30^\circ)^2 + (220\sin30^\circ)^2}\\ &= \sqrt{150^2 + 220^2 + 2\cdot150\cdot220\cos30^\circ} \end{align*}$$

(Used $\cos^2\theta+\sin^2\theta=1$)

Step3: Define displacement vectors for Plane 2

First leg: $\vec{u_1} = \langle 220, 0
angle$
Second leg: $150$ miles at $x^\circ$ south of east:
$\vec{u_2} = \langle 150\cos x^\circ, -150\sin x^\circ
angle$
Total displacement $\vec{D_2} = \vec{u_1}+\vec{u_2} = \langle 220 + 150\cos x^\circ, 0 - 150\sin x^\circ
angle$

Step4: Calculate magnitude for Plane 2

$$\begin{align*} |\vec{D_2}| &= \sqrt{(220 + 150\cos x^\circ)^2 + (-150\sin x^\circ)^2}\\ &= \sqrt{220^2 + 2\cdot220\cdot150\cos x^\circ + (150\cos x^\circ)^2 + (150\sin x^\circ)^2}\\ &= \sqrt{150^2 + 220^2 + 2\cdot150\cdot220\cos x^\circ} \end{align*}$$

Step5: Compare the two magnitudes

The only differing term is $\cos30^\circ$ vs $\cos x^\circ$. Since $x<30^\circ$, and $\cos\theta$ decreases as $\theta$ increases in $0^\circ$ to $90^\circ$, so $\cos x^\circ > \cos30^\circ$.
Thus $|\vec{D_2}| > |\vec{D_1}|$.

Answer:

The second plane is farther from the airstrip.