PLEASE ANSWER ASAP!!!!!!!!!!!!!!!!!!!!!!!

a cross country runner runs 3km due east, 5km due south and 8km south west. what is the runners final displacement and what direction in true bearing form?

PLEASE CAN YOU EXPLAIN IT STEP BY STEP

The north-south component of the displacement is

Y = -5 -8 cos 45 = -10.65 km

The west-east component of the displacement is
X = -3 +8 cos45 = 2.657 km

Final displacement = sqrt[(10.65)^2 + 2.657^2] = 10.976 km

Bearing = tan^-1 2.657/10.65
= 14 degrees S of E
= 104 clockwise from north

Sure, I can help you step by step to find the runner's final displacement and direction.

Step 1: Plot the distances traveled on a coordinate plane. Start by drawing a horizontal line (representing due east) of length 3 km from the origin (0,0). Then draw a vertical line (representing due south) of length 5 km from the end of the first line (3,0). Lastly, draw a diagonal line (representing south-west) of length 8 km from the end of the second line (3,-5).

Step 2: Connect the starting point (0,0) to the end point of the diagonal line to close the triangle.

Step 3: Find the coordinates of the end point of the diagonal line. Since the diagonal line represents a southwest direction, we need to go 8 km in that direction. The end point's coordinates will be (-5, -13).

Step 4: Calculate the displacement vector, which is the straight line distance between the starting point and the end point. Use the distance formula (Pythagorean theorem) to find the length of this displacement vector. The distance formula is:

distance = √[(x2 - x1)^2 + (y2 - y1)^2]

You have the coordinates of the starting point (x1, y1) = (0, 0) and the end point (x2, y2) = (-5, -13). Plugging in these values, we get:

distance = √[(-5 - 0)^2 + (-13 - 0)^2]
= √[25 + 169]
= √194
≈ 13.93 km (rounded to two decimal places)

So, the runner's final displacement is approximately 13.93 km.

Step 5: Find the direction (angle) of the displacement. Start by locating the angle that the displacement vector makes with the positive x-axis. To calculate this angle, you can use the inverse tangent function (arctan) with the slope of the displacement vector:

angle = arctan[(y2 - y1) / (x2 - x1)]

Plugging in the values, we get:

angle = arctan[(-13 - 0) / (-5 - 0)]
= arctan[-13 / -5]
≈ arctan[2.6]
≈ 69.4° (rounded to one decimal place)

Step 6: Convert the angle to true bearing form. In true bearing form, the direction is expressed as the angle measured clockwise from the north direction. Since the angle we obtained in Step 5 is measured counterclockwise from the positive x-axis, we need to subtract it from 360 degrees to get the true bearing:

true bearing = 360° - angle
= 360° - 69.4°
≈ 290.6° (rounded to one decimal place)

So, the runner's final displacement is approximately 13.93 km in a direction of 290.6° true bearing.

I hope this explanation helps! Let me know if you have any further questions.