Answer:
The magnitude of the helicopter's resultant vector is approximately 38.34 miles.
Explanation:
To find the magnitude of the helicopter's resultant vector, you can use vector addition. In this case, you have two displacement vectors: one from flying 11 miles south and another from turning 66 degrees east of south and traveling 42 miles.
First, let's represent these vectors:
The displacement vector from flying 11 miles south can be represented as:
Magnitude: 11 miles
Direction: South (which is in the negative y-direction)
The displacement vector from turning 66 degrees east of south and traveling 42 miles can be represented as:
Magnitude: 42 miles
Direction: 66 degrees east of south
Now, we'll break down the second vector into its x (east) and y (south) components. To do this, we'll use trigonometry. Since the helicopter is traveling east and south, we need to calculate the eastward and southward components of this vector.
Eastward Component:
Magnitude of the eastward component = 42 miles * cos(66 degrees)
Magnitude of the eastward component ≈ 42 miles * 0.4067 ≈ 17.04 miles (rounded to two decimal places)
Southward Component:
Magnitude of the southward component = 42 miles * sin(66 degrees)
Magnitude of the southward component ≈ 42 miles * 0.9135 ≈ 38.31 miles (rounded to two decimal places)
Now, we can add up the components of the two vectors:
The eastward component of the second vector is added to zero because the first vector is purely south, so it has no eastward component.
The southward component of the second vector is added to the first vector's southward component.
Resultant Magnitude = √(Eastward Component^2 + Southward Component^2)
Resultant Magnitude = √(0^2 + 38.31^2)
Resultant Magnitude ≈ √(1471.6761) ≈ 38.34 miles (rounded to two decimal places)