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?
To find the total displacement of the salesperson from the office, we need to calculate both the north-south and east-west components of the displacement and then combine them.
First, let's calculate the north-south displacement. The salesperson drives 26 km north in a straight line. Since the direction is north, this means the north-south displacement is positive. Therefore, the north-south displacement is +26 km.
Next, let's calculate the east-west displacement. The salesperson drives 62 km in a direction of 60 degrees from the north. To find the east-west displacement, we need to find the horizontal component of this distance. We can calculate this using trigonometry.
The east-west displacement (x) can be calculated using the formula:
x = distance * cos(angle)
In this case, the distance is 62 km and the angle is 60 degrees. Substituting these values into the formula, we have:
x = 62 km * cos(60 degrees)
Using a calculator, we can find that cos(60 degrees) is equal to 0.5. Substituting this value into the formula, we have:
x = 62 km * 0.5
x = 31 km
Since the direction is to the east, the east-west displacement is positive. Therefore, the east-west displacement is +31 km.
Now, we can combine the north-south and east-west displacements to find the total displacement. Since displacement is a vector quantity, we need to use vector addition. We can use the Pythagorean theorem to find the magnitude of the combined displacement.
The total displacement (D) can be calculated using the formula:
D = sqrt((north-south displacement)^2 + (east-west displacement)^2)
Substituting the values we found earlier, we have:
D = sqrt((26 km)^2 + (31 km)^2)
Using a calculator, we can find that the square root of (26 km)^2 + (31 km)^2 is approximately 40.3 km.
Therefore, the total displacement of the salesperson from the office is approximately 40.3 km.