Problem 37 - Trigonometry

Question

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. Heading and Distance Two planes take off at the same time from an airport. The first plane is flying at 246 miles per hour on a course of \(135.0^{\circ}\). The second plane is flying in the direction \(175.0^{\circ}\) at 357 miles per hour. Assuming there are no wind currents blowing, how far apart are they after 2 hours?

Step-by-Step Solution

Answer

The planes are about 462.67 miles apart after 2 hours.

Step 1: Understand the Problem

Two planes are traveling on different courses from the same starting point. Plane 1 travels at 246 mph at a course of \(135.0^{\circ}\), and Plane 2 travels at 357 mph at \(175.0^{\circ}\). We need to find the distance between them after 2 hours.

Step 2: Calculate the Distance Traveled by Each Plane

Distance is calculated as speed multiplied by time. Plane 1 travels:\[ D_1 = 246 \text{ mph} \times 2 \text{ hours} = 492 \text{ miles} \]Plane 2 travels:\[ D_2 = 357 \text{ mph} \times 2 \text{ hours} = 714 \text{ miles} \]

Step 3: Convert Courses to Cartesian Coordinates

Convert the courses to x,y-coordinates. Using angle \(\theta\), where \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\):For Plane 1:\[ x_1 = 492 \cos(135^{\circ}) = -348.48 \text{ miles} \]\[ y_1 = 492 \sin(135^{\circ}) = 348.48 \text{ miles} \]For Plane 2:\[ x_2 = 714 \cos(175^{\circ}) = -712.03 \text{ miles} \]\[ y_2 = 714 \sin(175^{\circ}) = 62.28 \text{ miles} \]

Step 4: Calculate Distance Between the Two Planes

Use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute the values:\[ d = \sqrt{((-712.03) - (-348.48))^2 + ((62.28) - (348.48))^2} \]\[ = \sqrt{(-363.55)^2 + (-286.2)^2} \]\[ = \sqrt{132165.60 + 81993.44} \]\[ = \sqrt{214159.04} \]\[ \approx 462.67 \text{ miles} \]

Step 5: Conclude the Results

After 2 hours, the two planes are approximately 462.67 miles apart.

Key Concepts

Bearing and Heading

When navigating, understanding the concepts of bearing and heading is crucial. Bearing is a method for specifying the direction of travel, usually using degrees. It's measured from the north in a clockwise direction. Heading, on the other hand, refers to the direction the aircraft is pointing—much like aiming the nose of a plane. It is also given in degrees relative to north.
For this exercise:

• Plane 1 has a heading of \(135.0^{\circ}\), meaning it is moving southeast from the starting point.
• Plane 2 follows a heading of \(175.0^{\circ}\), which is almost directly south.

Understanding these concepts helps visualize the planes' trajectories from their common origin at the airport, setting up our initial problem scenario.

Distance Calculation

Calculating how far each plane travels is essential. The formula to determine distance is: \[ D = ext{Speed} \times ext{Time} \] Since the planes fly for 2 hours, we multiply their speeds by this time.
For Plane 1, flying at 246 mph:

• The distance covered is \(246 \times 2 = 492\) miles.

For Plane 2, flying at 357 mph:

• The distance covered is \(357 \times 2 = 714\) miles.

These distances are useful for determining the position of each plane on a coordinate grid.

Coordinate Conversion

To find the positions of the planes on a 2D plane, we convert their directions into Cartesian coordinates. This involves using sine and cosine functions from trigonometry:

• For the x-component, use \(x = r \cos(\theta)\)
• For the y-component, \(y = r \sin(\theta)\)

For Plane 1, given \(135^{\circ}\), and knowing the total distance \(r = 492\):\[ (x_1, y_1) = (492 \cos(135^{\circ}), 492 \sin(135^{\circ})) = (-348.48, 348.48) \] For Plane 2 with \(175^{\circ}\) and \(r = 714\):\[ (x_2, y_2) = (714 \cos(175^{\circ}), 714 \sin(175^{\circ})) = (-712.03, 62.28) \] These coordinates will allow us to use the distance formula to calculate the separation between the planes.

Distance Formula in Trigonometry

The distance formula allows us to determine the straight-line distance between two points in a plane. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between them is calculated by:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates for the two planes:

• Plane 1 at \((-348.48, 348.48)\)
• Plane 2 at \((-712.03, 62.28)\)

Calculate:\[ d = \sqrt{((-712.03) - (-348.48))^2 + ((62.28) - (348.48))^2} \] This results in a distance of approximately 462.67 miles, indicating how far apart the planes are after two hours.