Tammy leaves the office, drives 51 km due north, then turns onto a second highway and continues in a direction of 31◦ north of east for 82 km. What is her total displacement from the office? Answer in units of km.
At what angle is her displacement? (Consider east to be 0◦ and north 90◦.) Answer in units of ◦.
A. Disp. = 51km[90o] + 82km[31o].
Disp. = 51i + (70.29+42.23i) = 70.29 + 93.2i. = sqrt(X^2+Y^2).
B. Tan A = Y/X.
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To find Tammy's total displacement from the office, we need to find the vector sum of her two displacements.
1. First, we calculate the northward displacement.
Tammy drives 51 km due north. Since this is a straight line, the northward component is the full distance.
Northward displacement = 51 km.
2. Next, we calculate the eastward displacement.
Tammy drives 82 km at an angle of 31° north of east.
To find the eastward component, we can use trigonometry.
Eastward displacement = 82 km * cos(31°).
3. Now, we can calculate the total displacement using the pythagorean theorem.
Total displacement = √(Northward displacement^2 + Eastward displacement^2).
Plugging in the values we have:
Total displacement = √((51 km)^2 + (82 km * cos(31°))^2).
Evaluating the expression gives us the total displacement.
To find the angle of her displacement, we can use the inverse tangent function.
Angle of displacement = arctan(Eastward displacement / Northward displacement).
Plugging in the values we have:
Angle of displacement = arctan((82 km * cos(31°)) / 51 km).
Evaluating the expression gives us the angle of her displacement.
To find Tammy's total displacement from the office, we need to find the horizontal and vertical components of her displacement separately and then combine them.
First, let's find the horizontal component. Tammy drives 82 km in a direction 31° north of east. To find the horizontal component, we use the cosine function:
Horizontal component = 82 km × cos(31°)
Next, let's find the vertical component. Tammy drives 51 km due north. To find the vertical component, we use the sine function:
Vertical component = 51 km × sin(90°)
Since she is driving due north, the angle is 90°.
Now, we can find the total displacement by adding the horizontal and vertical components:
Total displacement = √(Horizontal component² + Vertical component²)
The angle of the displacement can be found using the inverse tangent function:
Angle = tan^(-1)(Vertical component / Horizontal component)
Let's calculate the values:
Horizontal component = 82 km × cos(31°) ≈ 70.157 km
Vertical component = 51 km × sin(90°) = 51 km
Total displacement = √(70.157 km² + 51 km²) ≈ 87.684 km
Angle = tan^(-1)(51 km / 70.157 km) ≈ 35.696°
Therefore, Tammy's total displacement from the office is approximately 87.684 km, and the angle of her displacement is approximately 35.696°.