tammy leaves the office drives 50 km due north, then turns onto a second highway and continues in a direction of 32 degrees north of east for 93 km. At what angle is her displacement? (consider east to be 0 degrees and north 90 degrees)
50km N = (0,50)
93km E32°N = (78.9,49.3)
total displacement: (78.9,99.3)
arctan 99.3/78.9 = 51.5° = E51.5°N
To determine the angle of Tammy's displacement, we can use trigonometry. We need to find the angle between the east direction (0 degrees) and the direction of Tammy's displacement.
We can break Tammy's displacement down into horizontal and vertical components. The horizontal component is the distance she travels in the east direction, and the vertical component is the distance she travels in the north direction.
Horizontal component = 93 km * cos(32 degrees)
Vertical component = 93 km * sin(32 degrees)
To find the angle of displacement, we can use the inverse tangent (arctan) function:
Displacement angle = arctan(vertical component / horizontal component)
Let's calculate the angle:
Horizontal component = 93 km * cos(32 degrees) ≈ 78.833 km
Vertical component = 93 km * sin(32 degrees) ≈ 49.646 km
Displacement angle = arctan(49.646 km / 78.833 km) ≈ 32.77 degrees
Therefore, Tammy's displacement angle is approximately 32.77 degrees.
To find the angle of Tammy's displacement, we can use trigonometry and vector addition. Let's break down Tammy's displacement into its components.
First, she drives 50 km due north, which means she has moved directly in the positive y-direction. This component can be represented as (0, 50).
Then, she turns onto a second highway and continues in a direction of 32 degrees north of east for 93 km. We can break down this displacement into x and y components using trigonometry.
The horizontal component in the x-direction is given by 93 km * cos(32 degrees), and the vertical component in the y-direction is given by 93 km * sin(32 degrees).
Therefore, the x-component is approximately 79.822 km, and the y-component is approximately 49.512 km.
Now we can find the total displacement by adding up the corresponding components. Adding the x-components together gives us 79.822 km, and adding the y-components together gives us 99.512 km.
Now, we can find the angle of Tammy's displacement using the inverse tangent function. The formula to find the angle is:
angle = arctan(y-component / x-component)
Plugging in the values we obtained, we get:
angle = arctan(99.512 km / 79.822 km)
Using a calculator, we find that the angle is approximately 51.05 degrees.
Therefore, Tammy's displacement is at an angle of approximately 51.05 degrees.