A salesperson leaves the office and drives 26 km north along a straight highway. A turn is made onto a highway that leads in a direction of 60 degrees. The driver continues on the highway for a distance of 62 km and then stops. What is the total displacement of the salesperson from the office?
D = 26km[90o] + 62km[60o].
X = 62*Cos60 = 31km.
Y = 26 + 62*sin60 = 79.7 km.
Q1.
Tan A = Y/X = 79.7/31 = 2.57076.
A = 68.7o = Direction.
D = Y/sin A = 79.7/sin68.7 = 85.5 km[68.7o].
To find the total displacement of the salesperson from the office, we need to calculate the horizontal and vertical components separately and then combine them.
Step 1: Calculate the horizontal component of displacement
The salesperson drove 26 km north, so the horizontal component of displacement is zero.
Step 2: Calculate the vertical component of displacement
The salesperson traveled 62 km in a direction of 60 degrees. To find the vertical component, we can use trigonometry. The formula for calculating a side of a right-angled triangle is given by: side = hypotenuse * sin(angle).
Vertical component = 62 km * sin(60 degrees)
Vertical component = 62 km * 0.866
Vertical component ≈ 53.77 km
Step 3: Calculate the total displacement
Since the horizontal component is zero, the total displacement is equal to the vertical component.
Total displacement ≈ 53.77 km
Therefore, the total displacement of the salesperson from the office is approximately 53.77 km.
To find the total displacement of the salesperson from the office, we need to determine the straight-line distance between the starting point and the final destination. This can be found by using vector addition.
First, we need to break down the given information into x and y components. Let's assume the north direction is the positive y-axis, and the east direction is the positive x-axis.
1. The salesperson drives 26 km north. This can be represented as a vector in the y direction: 26 km in the positive y direction.
2. The salesperson then turns onto a highway that leads in a direction of 60 degrees. Since the initial north direction forms a 90-degree angle with the east direction, the 60-degree angle indicates a northeast direction. We can break down this vector into x and y components using trigonometry.
The x component: 62 km * cos(60 degrees) = 62 km * 0.5 = 31 km (positive x direction).
The y component: 62 km * sin(60 degrees) = 62 km * √3/2 ≈ 53.84 km (positive y direction).
Now we can add the vectors together:
Total x component = 0 km + 31 km = 31 km
Total y component = 26 km + 53.84 km = 79.84 km
Using the Pythagorean theorem, we can find the magnitude (total displacement):
Magnitude = √(31 km)^2 + (79.84 km)^2 ≈ √961 + 6,371.61 ≈ √7,332.61 ≈ 85.64 km
Therefore, the total displacement of the salesperson from the office is approximately 85.64 km.