as she pick up her riders a bus driver traverses four successive displacements represented by the expression (-6.30 b)i-(4.00 b cos 40)i-(4.00 sin 40)j+(3.00 b cos 50)i-(3.00 b sin 50)j-(5.00 b)j here b represents one city block,a convenient unit of distance of uniform size; i is east; and j is north. the displacements at 40 degree and 50 degree represent travel on roadways in the city that are at these angles to the main east-west and north-south streets.(a)draw a map of the successive displacements(b)what total distance did she travel? (c)compute the magnitude and direction of her total displacements.

*Please explain how you got to your answers/ how you found what to add together, etc*

Well, you will have to draw the map

for displacements add x components and y components separately and for distances:
find each and add
move 1
distance = 6.3
displacement east = -6.3 i
displacement north = 0 j

move 2
distance = 4
displacement east = -4 cos 40 = -3.06 i
displacement north= -4 sin 40 = -2.57 j

move 3
distance = 3
displacement east = 3 cos 50 = 1.93 i
displacement north=-3 sin 50 =-2.30 j

move 4
distance = 5
displacement east = 0 i
displacement north=-5 j

then total distance = 6.3+4+3+5

disp east = -6.3-3.06+1.93+0 =
disp North = 0 -2.57 -2.30 -5

tan angle = disp east/disp north
it will be in quadrant 3 (south west) here so that angle is south of west

oh, put all those in units of b, city blocks :)

To solve this problem, we can break down each displacement vector into its components and then add them together to find the total displacement.

(a) Drawing the map of the successive displacements:
Let's represent each displacement vector by an arrow in a coordinate system. The unit of distance is one city block, and 'i' represents east (+x direction), while 'j' represents north (+y direction).

(-6.30b)i: This vector points 6.30 blocks to the west (-x direction).
(-4.00b*cos(40))i: This vector points 4.00*cos(40) blocks to the east (+x direction).
(-4.00*sin(40))j: This vector points 4.00*sin(40) blocks to the south (-y direction).
(3.00b*cos(50))i: This vector points 3.00*cos(50) blocks to the east (+x direction).
(-3.00b*sin(50))j: This vector points 3.00*sin(50) blocks to the south (-y direction).
(-5.00b)j: This vector points 5.00 blocks to the north (+y direction).

Now, let's draw these vectors one after another starting from the origin, and then connect the end point of each vector to the start point of the next vector.

(-4.00*sin(40))j
|
|
(-6.30b)i--(-4.00b*cos(40))i--(3.00b*cos(50))i
|
|
(-5.00b)j

(b) Finding the total distance traveled:
The total distance traveled is simply the sum of the magnitudes of each displacement vector. We can calculate the magnitude of a vector using the Pythagorean theorem.

Magnitude of (-6.30b)i = √((-6.30b)^2) = 6.30b
Magnitude of (-4.00b*cos(40))i = √((-4.00b*cos(40))^2) = 4.00b*cos(40)
Magnitude of (-4.00*sin(40))j = √((-4.00*sin(40))^2) = 4.00*sin(40)
Magnitude of (3.00b*cos(50))i = √((3.00b*cos(50))^2) = 3.00b*cos(50)
Magnitude of (-3.00b*sin(50))j = √((-3.00b*sin(50))^2) = 3.00b*sin(50)
Magnitude of (-5.00b)j = √((-5.00b)^2) = 5.00b

Total distance = 6.30b + 4.00b*cos(40) + 4.00*sin(40) + 3.00b*cos(50) + 3.00*sin(50) + 5.00b

(c) Computing the magnitude and direction of the total displacement:
To find the magnitude of the total displacement, we use the Pythagorean theorem.

Magnitude of the total displacement = √((total distance)^2) = √[(6.30b)^2 + (4.00b*cos(40))^2 + (4.00*sin(40))^2 + (3.00b*cos(50))^2 + (3.00*sin(50))^2 + (5.00b)^2]

To find the direction of the total displacement, we can use trigonometry. We can determine the angle between the x-axis and the total displacement vector using arctan(y/x), where y and x are the northward and eastward components, respectively.

Angle = arctan[(sum of northward components)/(sum of eastward components)]
Angle = arctan[((-4.00*sin(40)) + (-3.00b*sin(50)) + (5.00b))/(6.30b + (4.00b*cos(40)) + (3.00b*cos(50)))]

Simplify the expression to get the value of the angle.

I hope this explanation helps you understand how to solve the problem. Remember to substitute the value of 'b' with a specific number to obtain numerical answers.