A plane flies due east from point P (30 N, 40 W) for 4500 km to point Q. Find the longitude of Q.

Question

ScanSolve AI Tutor · Accepted Answer

Here's how to find the longitude of Q:

First, let's define the given information:

Point P: $(30^\circ N, 40^\circ W)$
Distance PQ: $4500 \text{ km}$
Flight direction: Due east, on the same latitude. This means the latitude of Q is also $30^\circ N$ .
We will use the Earth's radius $R = 6371 \text{ km}$ .

The distance along a parallel of latitude is given by the formula:

L = \frac{\theta}{360^\circ} \times 2\pi R \cos \alpha

where:

$L$ is the distance along the parallel ( $4500 \text{ km}$ )
$\theta$ is the difference in longitude in degrees
$R$ is the radius of the Earth ( $6371 \text{ km}$ )
$\alpha$ is the latitude ( $30^\circ$ )

Draw an appropriate diagram.
A diagram showing the Earth, the Equator, the $30^\circ N$ parallel of latitude, and the positions of P and Q.
\begin{tikzpicture}[scale=0.8] % Earth \draw (0,0) circle (3cm); % Equator \draw (-3,0) -- (3,0) node[right] {Equator}; % Prime Meridian (simplified) \draw (0,-3) -- (0,3) node[above] {PM}; % Latitude 30N \draw[dashed] (0, 3*sin(30)) ellipse (3*cos(30) and 0.5cm); % Ellipse for 30N parallel \node at (3*cos(30), 3*sin(30)) [right] {$30^\circ N$}; % Point P (40W) \coordinate (P) at ({3*cos(30)*cos(40)}, {3*cos(30)*sin(40)}); \fill (P) circle (1.5pt) node[below left] {P ($40^\circ W$)}; \draw[dashed] (0,0) -- (P); % Meridian line for P % Point Q (East of P) % We'll calculate the angle first, then place Q % For now, just indicate direction \node at (0, 3*sin(30)) {}; % Center of the parallel \draw[->, thick] (P) arc[start angle={180-40}, end angle={180-40+46.7}, radius={3*cos(30)}] node[midway, above] {$4500 \text{ km}$}; % Calculate Q's position for the diagram \pgfmathsetmacro{\theta_deg}{46.7} \pgfmathsetmacro{\lambda_Q_deg}{-40 + \theta_deg} \coordinate (Q) at ({3*cos(30)*cos(-\lambda_Q_deg)}, {3*cos(30)*sin(-\lambda_Q_deg)}); \fill (Q) circle (1.5pt) node[below right] {Q ($\lambda_Q$)}; \draw[dashed] (0,0) -- (Q); % Meridian line for Q % Angle theta at the pole (simplified representation) \draw[<->] ({3*cos(30)*cos(40)*0.8}, {3*cos(30)*sin(40)*0.8}) arc[start angle={180-40}, end angle={180-40+46.7}, radius={3*cos(30)*0.8}]; \node at ({3*cos(30)*cos(40-46.7/2)*0.9}, {3*cos(30)*sin(40-46.7/2)*0.9}) {$\theta$}; \end{tikzpicture}
Substitute the known values into the formula.
$4500 = \frac{\theta}{360^\circ} \times 2\pi (6371 \text{ km}) \cos(30^\circ)$
Calculate $\cos(30^\circ)$ .
$\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866025$
Solve for $\theta$ .
$4500 = \frac{\theta}{360^\circ} \times 2\pi (6371) (0.866025)$ $4500 = \theta \times \frac{2\pi \times 6371 \times 0.866025}{360}$ $4500 = \theta \times \frac{34680.16}{360}$ $4500 = \theta \times 96.3338$ $\theta = \frac{4500}{96.3338}$ $\theta \approx 46.71^\circ$
This is the angular difference in longitude between P and Q.
Determine the longitude of Q.
Point P is at $40^\circ W$ . The aeroplane flies due east, so the longitude increases (moves towards $0^\circ$ and then into the Eastern hemisphere).
Longitude of Q = Longitude of P + $\theta$ (when moving east)
Longitude of Q = $40^\circ W - 46.71^\circ$ (treating West as negative and East as positive, or simply moving $46.71^\circ$ east from $40^\circ W$ )

Starting at $40^\circ W$ , moving $40^\circ$ east brings us to the Prime Meridian ( $0^\circ$ ).
The remaining distance to travel east is $46.71^\circ - 40^\circ = 6.71^\circ$ .
Since we crossed the Prime Meridian and continued east, the longitude of Q is $6.71^\circ E$ .
Round to the nearest degree.
$6.71^\circ E \approx 7^\circ E$ .

The longitude of Q is $\text{7}^\circ E$ .

A plane flies due east from point P (30 N, 40 W) for 4500 km to point Q. Find the longitude of Q.

Answer

Step-by-step solution

Draw an appropriate diagram.

Substitute the known values into the formula.

Calculate $\cos(30^\circ)$ .

Solve for $\theta$ .

Determine the longitude of Q.

Round to the nearest degree.

Try one yourself

Still stuck on this one?

More Geography Questions

A plane flies due east from point P (30 N, 40 W) for 4500 km to point Q. Find the longitude of Q.

Answer

Step-by-step solution

Draw an appropriate diagram.

Substitute the known values into the formula.

Calculate cos⁡(30∘)\cos(30^\circ)cos(30∘).

Solve for θ\thetaθ.

Determine the longitude of Q.

Round to the nearest degree.

Try one yourself

Still stuck on this one?

More Geography Questions

Calculate $\cos(30^\circ)$ .

Solve for $\theta$ .