Using Parametric equations and vectors consider the following scenario:
Starting from an airport, an airplane flies 225 miles northwest, then 150 miles south-west.
Collaborate with your peers to answer the following:
Draw a graph or figure to represent this situation. (This graph must be digital on Desmos or equivelant not handdrawn.)
Describe how the concepts from this module can be applied in this case.
How far, in miles, from the airport is the plane?
Provide another example of a scenario that involves the same concept.
luckily NW is perpendicular to SW so d = sqrt (225^2 + 150^2)
northwest in standard trig notation ---- 135°
southwest in standard trig notation ---- 225°
(x,y) = 225(cos135,sin135) + 150(cos225,sin225)
= 225(-√2/2, √2/2) + 150(-√2/2, -√2/2)
= (-225√2/2 , 225√2/2) + (-150√2/2 , -150√2/2)
= (-375√2/2 , 75√2/2)
distance from airport
= √( (-375√2/2)^2 + (75√2/2)^2) = appr 270.4 miles
Unfortunately, as a text-based AI, I am unable to draw digital graphs on platforms like Desmos. However, I can describe the graph and provide step-by-step instructions on how you can draw it on Desmos yourself.
To draw the graph representing the airplane's flight path, you can follow these steps on Desmos:
1. Open Desmos on your preferred browser.
2. On the left-hand side, you will see an input bar where you can type equations or expressions.
3. In the input bar, type the following equation for the airplane's first leg flying 225 miles northwest: x = -225t, y = 225t, where t is the parameter representing time.
4. After typing the equation, hit enter or click on the "+" button next to the input bar to add a new equation.
5. Create a new equation for the second leg of the flight, where the airplane flies 150 miles southwest. You can use another parameter, such as t2, to represent time for this leg of the flight. The equation will be x = -150t2, y = -150t2.
6. Hit enter or click on the "+" button to add the equation to the graph.
7. You should now see two lines representing the flight path of the airplane. The first line will start at the origin (0, 0) and move northwest, while the second line will continue from the end of the first line and move southwest.
By applying the concepts from this module, you can use parametric equations and vectors to represent the position and displacement of the airplane relative to the airport. The parametric equations x = -225t, y = 225t represent the northwest leg of the flight, where the x and y coordinates correspond to the airplane's position at a given time. Similarly, the second set of parametric equations x = -150t2, y = -150t2 represent the southwest leg of the flight. By using vectors, you can determine the total displacement of the airplane by finding the difference between the final position and the initial position.
To find the distance, in miles, from the airport to the plane, you can calculate the magnitude of the displacement vector using the Pythagorean theorem. The displacement vector is the difference between the final position and the initial position.
In this case, the initial position is (0, 0) since the airplane starts from the airport. The final position is the endpoint of the southwest leg of the flight. To calculate the distance, you can substitute the values of x and y from the southwest leg equations into the Pythagorean theorem formula:
Distance = √(x^2 + y^2)
= √((-150t2)^2 + (-150t2)^2)
= √(2 * (-150t2)^2)
= √(2 * 22500t2^2)
= 150t2√2
Therefore, the distance from the airport to the plane is given by 150t2√2 miles, where t2 represents the time taken for the southwest leg of the flight.
To draw a graph representing this situation, we can use parametric equations to track the movement of the airplane. Let's use the origin as the starting point, and let x and y represent the position of the airplane.
The movement of the airplane can be defined by the following parametric equations:
x = 225cos(45°) + 150cos(225°)
y = 225sin(45°) + 150sin(225°)
Now let's plot these equations on a graphing tool like Desmos or any other equivalent online graphing calculator. Simply enter the equations into the graphing tool, and it will plot the path of the airplane.
To apply the concepts from this module, we can use vectors to represent the movements of the airplane. For example, the first movement of 225 miles northwest can be represented as a vector [225cos(45°), 225sin(45°)]. Similarly, the second movement of 150 miles southwest can be represented as a vector [150cos(225°), 150sin(225°)]. By summing these vectors, we can find the total displacement of the airplane from the starting point.
To find how far the plane is from the airport, we can calculate the magnitude of the total displacement vector. Using the Pythagorean theorem, the magnitude can be calculated as the square root of the sum of the squares of the x and y components:
Distance = sqrt((225cos(45°) + 150cos(225°))^2 + (225sin(45°) + 150sin(225°))^2)
Evaluate the above expression to get the distance in miles.
Another example that involves the same concept is a boat navigating through a river. Suppose a boat is moving at a speed of 15 mph directly across a river flowing at a speed of 5 mph. We can represent the boat's velocity vector as [15, 0] (since it's moving directly across the river) and the river's velocity vector as [0, 5] (since it's flowing the opposite way). By summing these vectors, we can find the resultant velocity vector of the boat relative to the ground. The magnitude of this resultant vector will give us the boat's speed relative to the ground.