Tammy drives 34 km due north, then continues in a direction of 32◦ north of east for 99 km.What is her total displacement from the office?
displacement=34kmN+ 99*cos32 N + 99*sin32E
combine the first two terms, you will have total displacement.
To calculate Tammy's total displacement from the office, we need to determine both the magnitude (distance) and direction of her displacement.
First, let's visualize Tammy's two displacements:
1. The first displacement of 34 km due north can be represented as an arrow pointing directly upwards.
2. The second displacement of 99 km in a direction 32 degrees north of east can be represented as an arrow that deviates slightly from the east direction towards the north.
To find the resultant displacement, we can use vector addition. We will add these two displacements using the Pythagorean theorem to find the magnitude and trigonometry to find the direction.
Let's calculate the magnitude:
magnitude = √(34^2 + 99^2)
magnitude = √(1156 + 9801)
magnitude = √10957
magnitude ≈ 104.7 km (rounded to one decimal place)
Next, let's calculate the direction:
We have two angles, the angle of the first displacement (0 degrees) and the angle of the second displacement (32 degrees).
To find the resultant direction, we can use the formula:
tan(θ) = opposite/adjacent
We can take the tangent of (32 degrees) and solve for the adjacent side, which represents the eastward component of Tammy's displacement.
Eastward component = 99 km * cos(32°)
Eastward component ≈ 83.9 km (rounded to one decimal place)
Now, to find the northward component, we can take the sine of (32 degrees) and multiply it by the magnitude of the first displacement:
Northward component = 34 km * sin(32°)
Northward component ≈ 17.4 km (rounded to one decimal place)
To find the overall direction, we use the inverse tangent:
Resultant direction = arctan(Northward component / Eastward component)
Resultant direction ≈ arctan(17.4 km / 83.9 km)
Resultant direction ≈ 11.86 degrees (rounded to two decimal places)
Therefore, Tammy's total displacement from the office is approximately 104.7 km in a direction 11.86 degrees north of east.