Solving Convert & Vector Equations: Cartesian, Polar, Push Ball

[Apophis]

Stuck on these...


1.Convert the following equation to Cartesian coordinates r sin (angle) = -2.

2.Convert the rectangular equation y = -5x + 4 to a polar equation.

3.An aircraft going from Atlanta, GA to New York, NY on a bearing of S69oE is traveling at a speed of 430 miles per hour. The wind is blowing out of the north to south at a speed of 25 miles per hour. Find the ground speed and the plane's true bearing.

4.Two teams are playing push ball with a large 8 foot diameter ball. One team exerts a force represented by the vector a = 2 i + -5 j , and the other team exerts a force represented by the vector b = -8 i -3 j . Determine the combined force magnitude.

 

[whozum]

The translators from polar to cartesian and backwards are:

$$x = r cos(\theta)$$

$$y = r sin(\theta)$$

And from the above two you can see that

$$x^2 + y^2 = r^2$$

This is enough for #1 and #2

3. Vector addition, draw a vector with heading 69 degrees south of east and label its magnitude 430mph. Then add another vector to it with south heading at 25mph. The total heading and speed will be the sum of the two.

4. Same as #3, draw the two vectors and find the vector sum.

 

[thepatientmental]

1. To convert the equation to Cartesian coordinates, we can use the identities x = r cos(angle) and y = r sin(angle). Therefore, the Cartesian equation would be x = -2 cos(angle).

2. To convert the equation to polar coordinates, we can use the identities x = r cos(angle) and y = r sin(angle). Therefore, the polar equation would be r = -5x + 4 or r = -5(r cos(angle)) + 4. This can be simplified to r = -5r cos(angle) + 4 or r(1+5cos(angle)) = 4.

3. To find the ground speed, we can use the formula: ground speed = airspeed + wind speed. Therefore, the ground speed would be 430 + 25 = 455 miles per hour. To find the true bearing, we can use the formula: true bearing = bearing + wind direction. Therefore, the true bearing would be S69oE + 90o = S159oE.

4. To find the combined force magnitude, we can use the formula: magnitude = √(a^2 + b^2). Plugging in the given values, we get magnitude = √(2^2 + (-5)^2) + √((-8)^2 + (-3)^2) = √(4 + 25) + √(64 + 9) = √29 + √73. This cannot be simplified any further without knowing the values of i and j.