Write an equation in slope-intercept form of the line whose parametric equations are x= (1/2)t + (2/3) and y= t – (3/4).
t = y + 3/4
x = 1/2(y + 3/4) + 2/3
2x = y + 3/4 + 4/3
2x = y + 25/12
y = 2x - 25/12
A force F1 of 35 newtons pulls at an angle of 15 degrees north of due east. A force F2 of 75 newtons pulls at an angle of 55 degrees west of due south. Find the magnitude and direction of the resultant force.
43.8 N, 39.1 degrees west of due south.
To find the magnitude and direction of the resultant force, we first need to break down the forces F1 and F2 into their x and y components.
For Force F1:
The angle is given as 15 degrees north of due east. To find the x-component and y-component, we can use trigonometry.
The x-component (F1x) can be found using the cosine function: F1x = F1 * cos(angle)
The y-component (F1y) can be found using the sine function: F1y = F1 * sin(angle)
For Force F2:
The angle is given as 55 degrees west of due south. To find the x-component and y-component, we use trigonometry again.
The x-component (F2x) can be found using the cosine function: F2x = F2 * cos(angle)
The y-component (F2y) can be found using the sine function: F2y = F2 * sin(angle)
Now that we have the x and y components for both forces, we can add them to find the resultant force.
The x-component of the resultant force (Rx) is the sum of the x-components of F1 and F2: Rx = F1x + F2x
The y-component of the resultant force (Ry) is the sum of the y-components of F1 and F2: Ry = F1y + F2y
To find the magnitude of the resultant force (R), we can use the Pythagorean theorem: R = sqrt(Rx^2 + Ry^2)
To find the direction of the resultant force, we can use trigonometry again. The angle (θ) can be found using the arctan function: θ = atan(Ry / Rx)
Substituting the given values into the equations, we can calculate the magnitude and direction of the resultant force:
F1 = 35 newtons, angle = 15 degrees
F2 = 75 newtons, angle = 55 degrees
Using the formulas above, we find:
F1x = 35 * cos(15 degrees)
F1y = 35 * sin(15 degrees)
F2x = 75 * cos(55 degrees)
F2y = 75 * sin(55 degrees)
Rx = F1x + F2x
Ry = F1y + F2y
R = sqrt(Rx^2 + Ry^2)
θ = atan(Ry / Rx)
Solving these equations will give you the magnitude and direction of the resultant force