Question

A barge is being towed by two tugboats, using ropes with force vectors as shown.

Answer 1

The angle between the two ropes, rounded to the nearest degree, is
`\( 53^\circ \)`. Therefore, option B is correct

To find the angle between the two vectors
`\( F_1 \)` and
`\( F_2 \),` we will:

1. Calculate the dot product of
`\( F_1 \)` and
`\( F_2 \).`

2. Find the magnitudes (or lengths) of
`\( F_1 \)` and
`\( F_2 \).`

3. Use the dot product and magnitudes to calculate the cosine of the angle between the two vectors.

4. Use the arccosine function to find the angle itself.

5. Convert the angle from radians to degrees.

Given:

`\[ F_1 = (11,000, 5,000) \]`

`\[ F_2 = (14,500, -8,000) \]`

The dot product
`\( F_1 \cdot F_2 \)` is calculated as:

`\[ F_1 \cdot F_2 = (11,000 \cdot 14,500) + (5,000 \cdot -8,000) \]`

The magnitudes of
`\( F_1 \)` and
`\( F_2 \)` are calculated as:

`\[ |F_1| = √((11,000)^2 + (5,000)^2) \]`

`\[ |F_2| = √((14,500)^2 + (-8,000)^2) \]`

The cosine of the angle
`\( \theta \)` is:

`\[ \cos(\theta) = (F_1 \cdot F_2)/(|F_1| \cdot |F_2|) \]`

Finally, the angle
`\( \theta \)` is:

`\[ \theta = \arccos\left((F_1 \cdot F_2)/(|F_1| \cdot |F_2|)\right) \]`

Now we will calculate these values step by step:

1. Dot Product: The dot product of the vectors \( F_1 \) and \( F_2 \) is calculated by multiplying corresponding components and then summing those products:

`\[ \text{Dot Product} = F_1 \cdot F_2 = (11,000 \cdot 14,500) + (5,000 \cdot (-8,000)) \]`

`\[ \text{Dot Product} = 159,500,000 - 40,000,000 \]`

`\[ \text{Dot Product} = 119,500,000 \]`

2. **Magnitude of \( F_1 \)**: The magnitude (or length) of vector \( F_1 \) is calculated using the Pythagorean theorem for the vector components:

`\[ |F_1| = √(11,000^2 + 5,000^2) \]`

`\[ |F_1| = √(121,000,000 + 25,000,000) \]`

`\[ |F_1| = √(146,000,000) \]`

`\[ |F_1| \approx 12,083.05 \]` (rounded to two decimal places for clarity)

3. Magnitude of
`\( F_2 \)`: Similarly, the magnitude of vector
`\( F_2 \)` is:

`\[ |F_2| = √(14,500^2 + (-8,000)^2) \]`

`\[ |F_2| = √(210,250,000 + 64,000,000) \]`

`\[ |F_2| = √(274,250,000) \]`

`\[ |F_2| \approx 16,560.50 \]` (rounded to two decimal places for clarity)

4. Cosine of the Angle: The cosine of the angle between the vectors is the dot product divided by the product of the magnitudes:

`\[ \cos(\theta) = \frac{\text{Dot Product}} \]`

`\[ \cos(\theta) = (119,500,000)/(12,083.05 \cdot 16,560.50) \]`

`\[ \cos(\theta) \approx (119,500,000)/(200,139,302.5) \]`

`\[ \cos(\theta) \approx 0.5972 \]` (rounded to four decimal places for clarity)

5. Angle
`\( \theta \)`: To find the angle in radians, we use the arccosine function:

`\[ \theta = \arccos(\cos(\theta)) \]`

`\[ \theta \approx \arccos(0.5972) \]`

`\[ \theta \approx 0.9303 \]` (in radians, rounded to four decimal places)

6. Convert to Degrees: Finally, convert the angle from radians to degrees:

`\[ \theta \text{ in degrees} = \frac{\theta \text{ in radians}}{\pi} * 180^\circ \]`

`\[ \theta \text{ in degrees} \approx (0.9303)/(\pi) * 180^\circ \]`

`\[ \theta \text{ in degrees} \approx 53.33^\circ \]`

Rounded to the nearest degree, the angle is
`\( 53^\circ \).`

The angle between the two ropes, rounded to the nearest degree, is
`\( 53^\circ \)`.

Answer 2

From the question, we are given

`\begin{gathered} F_1=(11,000,5000) \\ F_2=(14,500,-8,000) \end{gathered}`

Now, to find the angle between the ropes, we will take the dot product of two vectors. This is given as

`\begin{gathered} a.b=|a||b|cos\theta \\ so,\text{ we have } \\ cos\theta=(a.b)/(|a||b|) \end{gathered}`

But writing the forces in vector form, we have

`\begin{gathered} F_1=11,000i+5000j \\ F_2=14,500i-8,000j \end{gathered}`

Applying the formula above, we have

`\begin{gathered} cos\theta=(a.b)/(|a||b|) \\ cos\theta=\frac{(11,000)(14,500)+(5000)(-8000)}{\sqrt{11,000^2+5000^2)√(14500^2+(-8000)^2)}} \\ cos\theta=(159,500,000-40,000,000)/(√(121,000,000+25,000,000)√(210,250,000+64,000,000)) \\ cos\theta=(119,500,000)/(√(146,000,000)√(274,250,000)) \\ cos\theta=(119,500,000)/(12,083.04579*16560.4952) \\ cos\theta=(119,500,000)/(200,101,221.806675) \\ cos\theta=0.5971977 \end{gathered}`

Now, theta becomes

`\begin{gathered} \theta=cos^(-1)0.5971977 \\ \theta=53.3305 \end{gathered}`

hence the answer is option B, 53 degrees