Question

a helicopter flies 11 miles south. it then turns 66 degrees east of south and travels an additional 42 miles. what is the magnitude of the helicopters resultant vector. 50 pts!!!!!

Answer 1

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)

Answer 2

38.85 miles

Explanation:

The magnitude of a vector is its length. It is calculated using the Pythagorean theorem, which states that the square of the magnitude of a vector is equal to the sum of the squares of its components.

In this case, the first vector has a magnitude of 11 miles and the second vector has a magnitude of 42 miles. The angle between the two vectors is 66°.

We can use the cosine rule to find the magnitude of the resultant vector.

The cosine rule states that:

`\sf c^2 = a^2 + b^2 - 2ab * cos(\theta)`

where:

• c is the magnitude of the resultant vector
• a is the magnitude of the first vector (11 miles)
• b is the magnitude of the second vector (42 miles)
• θ is the angle between the two vectors (66°)

In this case,

Substitute the value

`\sf c^2 = 11^2 + 42^2 - 2 * 11 times 42 * cos(66^\circ)`

`\sf c^2 = 121 + 1764 - 924 * 0.4067366431`

`\sf c^2 =1885 - 375.8246582`

`\sf c^2 = 15089.175342`

`\sf c=√( 15089.175342)`

`\sf c= 38.84810603`

`\sf c= 38.85 \textsf{in nearest hundred}`

Therefore, magnitude of the helicopter's resultant vector is 38.85 miles