Math Question: A plane is at {0,0} travelling North. It receives a directive to reach a coordinate, North-East at {X,Y}. The plane has a huge turning radius R, which is to say it can't turn "on a dime" some degrees clock-wise. It will therefore stay on a circle trajectory for some time & distance. Then it will be able to go fly straight towards its coordinate and reach it. a. How many degrees will it stay on its circle trajectory? b. With arc length = 2(pi)(R)(ang/360), how much distance is that? c. After that distance on the circle, where is it in Cartesian space? d. With Pythagorean distance = sqrt(dx^2 + dy^2), how much distance is left?

To solve this problem, let's break it down step by step. ### a. How many degrees will it stay on its circle trajectory? The plane starts at (0, 0) and heads North. It needs to turn towards the North-East direction to reach the coordinate (X, Y). The angle it needs to turn is the angle between the North direction and the line connecting the origin (0, 0) to the point (X, Y). Since the plane is initially traveling North (along the positive y-axis), the angle \(\theta\) can be found using the arctangent function: \[ \theta = \arctan\left(\frac{X}{Y}\right) \] This angle is measured from the positive y-axis. ### b. With arc length = \(2\pi R \left(\frac{\theta}{360}\right)\), how much distance is that? Given the radius \(R\) of the turning circle, the arc length \(L\) that the plane travels along the circle can be calculated as: \[ L = 2\pi R \left(\frac{\theta}{360}\right) \] where \(\theta\) is in degrees. We need to convert \(\theta\) from radians to degrees if we use it in the formula directly: \[ L = 2\pi R \left(\frac{\theta \cdot 180}{\pi \cdot 360}\right) = R\theta \] ### c. After that distance on the circle, where is it in Cartesian space? After traveling the arc length, the plane's coordinates can be found using the parametric equations for a circle. Given that it started at the origin (0, 0) and traveled an angle \(\theta\) counterclockwise, the new coordinates (x, y) on the circle of radius \(R\) are: \[ x = R \sin(\theta) \] \[ y = R \cos(\theta) \] where \(\theta\) is in radians. ### d. With Pythagorean distance = \(\sqrt{(dx)^2 + (dy)^2}\), how much distance is left? The distance remaining to reach the coordinate (X, Y) from the new position (x, y) can be found using the Pythagorean theorem: \[ \text{Remaining distance} = \sqrt{(X - x)^2 + (Y - y)^2} \] Let's compile these steps into a complete solution. 1. **Calculate the angle \(\theta\):** \[ \theta = \arctan\left(\frac{X}{Y}\right) \] 2. **Calculate the arc length \(L\):** \[ L = R \theta \] 3. **Find the coordinates after traveling the arc length:** \[ x = R \sin(\theta) \] \[ y = R \cos(\theta) \] 4. **Calculate the remaining distance to the target coordinate:** \[ \text{Remaining distance} = \sqrt{(X - x)^2 + (Y - y)^2} \] Let's use these formulas to calculate the values for given \(X, Y, R\). Please provide the values for \(X, Y,\) and \(R\), so we can proceed with the calculations.
Answered On 07/19/2024 21:13