Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Two ships leave a port at the same time. The first ship sails on a bearing of \(40^{\circ}\) at 18 knots (nautical miles per hour) and the second at a bearing of \(130^{\circ}\) at 26 knots. How far apart are they after 1.5 hours?

Step-by-Step Solution

Answer

The ships are approximately 47.5 nautical miles apart after 1.5 hours.

Step 1: Calculate Distance Travelled by Each Ship

To find out how far each ship has traveled in 1.5 hours, we multiply their speed by the time traveled.For the first ship: \[\text{Distance} = 18 \times 1.5 = 27 \text{ knots}\]For the second ship:\[\text{Distance} = 26 \times 1.5 = 39 \text{ knots}\]

Step 2: Determine Coordinate Positions

To use trigonometry, convert the movement of the ships into coordinate positions from the origin.The first ship:\[\begin{align*}\text{X-coordinate} &= 27 \cdot \cos(40^{\circ}) \\text{Y-coordinate} &= 27 \cdot \sin(40^{\circ})\end{align*}\]The second ship:\[\begin{align*}\text{X-coordinate} &= 39 \cdot \cos(130^{\circ}) \\text{Y-coordinate} &= 39 \cdot \sin(130^{\circ})\end{align*}\]Calculate these values using a calculator.

Step 3: Calculate the Straight-Line Distance Between the Two Ships

Use the distance formula to find the straight-line distance between the two ships with coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \):\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the values found in Step 2 into this formula to find the distance.

Step 4: Final Calculation

First, calculate the precise positions of each ship:\[\begin{align*}x_1 &= 27 \cdot \cos(40^{\circ}) \ &\approx 20.7 \y_1 &= 27 \cdot \sin(40^{\circ}) \ &\approx 17.4 \x_2 &= 39 \cdot \cos(130^{\circ}) \ &\approx -25.1 \y_2 &= 39 \cdot \sin(130^{\circ}) \ &\approx 29.9 \\end{align*}\]Now, substitute these into the distance formula:\[d = \sqrt{(20.7 - (-25.1))^2 + (17.4 - 29.9)^2} \]Calculate the result:\[d = \sqrt{45.8^2 + (-12.5)^2} \]\[d = \sqrt{2097.64 + 156.25} \]\[d \approx \sqrt{2253.89} \]\[d \approx 47.5 \text{ nautical miles}\]

Key Concepts

Distance Formula

The distance formula is a key concept in coordinate geometry, allowing us to calculate the straight-line distance between two points on a plane. It is derived from the Pythagorean theorem, where the difference in the x-coordinates of two points provides one side of a right triangle, and the difference in the y-coordinates provides the other. The hypotenuse, which is the distance between the points, is calculated using these differences.
In this exercise, the positions of the ships are converted into coordinates, and the formula is used as follows:

• Given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance \( d \) is calculated using: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• Here, the differences in x and y are squared, summed, and then we take the square root, providing the shortest distance between these two points.

This formula allows us to transform the abstract path of moving objects into tangible numbers, crucial for precise navigation tasks.

Bearing Angle Calculation

Understanding bearing angles is crucial for navigation, especially in maritime contexts. A bearing is a horizontal angle measured clockwise from a north baseline. This navigational method simplifies the process of determining direction and distance.
To calculate a bearing angle:

• Start with a north-south line or baseline. Directions are measured clockwise from this line.
• For example, in our scenario, one ship moves at a bearing of \( 40^\circ \) and the other at \( 130^\circ \), indicating their clockwise rotation from north.

Using trigonometric functions:

• Cosine for x-coordinate: involves multiplying the distance by the cosine of the bearing angle.
• Sine for y-coordinate: involves multiplying the distance by the sine of the bearing angle.

Bearings help translate directional navigation into coordinate geometry, aiding in precise sailing and positioning.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, integrates algebra and geometry. It revolutionizes solving spatial problems by using algebraic equations to represent geometric figures and positions in a coordinate plane.
In the problem at hand, coordinate geometry helps position the ships:

• Plane is defined with an origin point and coordinates (x, y) to locate positions.
• Using trigonometry, each ship's position becomes a point in the coordinate system, derived from their respective bearing angles and distances traveled.
• With ships, the x-coordinate is obtained using cosine, and the y-coordinate using sine of the bearing angle, multiplied by the distance.

Using coordinate geometry allows for precise and efficient calculations, proving invaluable in scenarios involving navigation and distances across the earth's surface.