A Plane Departs From An Airport At Coordinates 23
A plane departs from an airport at the coordinates 23 And
A plane departs from an airport at the coordinates (2,3) and heads towards another airport at the coordinates (10,8). Your task is to determine the equation of the flight path. Find the slope of the line representing the flight path using the given coordinates. Write the equation of the line in the slope-intercept form (y = mx + b). Explain the significance of each component in the equation in relation to the plane's flight path.
Ensure that your solution includes all the necessary calculations and a brief explanation of each step. Found the answer on this website but I'm not sure if it's correct.
Paper For Above instruction
The problem involves determining the equation of the flight path of a plane traveling between two specified coordinates: the starting point at (2, 3) and the destination at (10, 8). To achieve this, we need to find the line that passes through these two points, which represents the plane's flight route. This process involves calculating the slope of the line, deriving the equation in slope-intercept form (y = mx + b), and understanding the meaning of each component in the context of the aircraft's trajectory.
Calculating the Slope of the Flight Path
The first step is to compute the slope (m) of the line connecting the two points. The slope formula is given by:
m = (y₂ - y₁) / (x₂ - x₁)
where (x₁, y₁) = (2, 3) and (x₂, y₂) = (10, 8). Substituting these values:
m = (8 - 3) / (10 - 2) = 5 / 8
The slope of the flight path is therefore 5/8.
Deriving the Equation of the Line
Using the slope-intercept form y = mx + b, where m is the slope we just calculated, we now find the y-intercept (b). To do this, substitute one of the given points into the equation along with the slope. Let's use point (2, 3):
3 = (5/8) × 2 + b
Simplifying:
3 = (10/8) + b
3 = (5/4) + b
To find b, subtract 5/4 from both sides:
b = 3 - 5/4 = (12/4) - (5/4) = (7/4)
Thus, the equation of the flight path in slope-intercept form is:
y = (5/8)x + 7/4
Significance of Each Component
In the equation y = (5/8)x + 7/4:
- y represents the altitude or position of the plane at a given x-coordinate (which could relate to east-west position if x is longitude or another directional measure).
- m = 5/8 is the slope, indicating the rate of change of the y-coordinate with respect to x. It reflects the incline or descent of the flight path, with a positive slope indicating an ascent or northeast direction.
- b = 7/4 is the y-intercept, representing the initial position of the plane when x=0. Although not directly on the route between airports, it provides the starting point of the line's extension.
Overall, the equation models the direct linear path that the aircraft follows between the two airports, with the slope indicating the inclination of the trajectory and the intercept indicating its initial position.
Conclusion
The calculated line equation, y = (5/8)x + 7/4, effectively describes the plane's flight path between the departure point at (2,3) and the destination at (10,8). Such a mathematical representation allows for analysis of the aircraft's course and could serve as a basis for navigation or further spatial analysis. Accurate determination of this line also aids in understanding the spatial relationship between the two points and planning for optimal flight routes.
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