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To solve this problem, we need to calculate the length of the vector \( D \) in kilometers. We are given a four-sided plot of land, and sides \( A \), \( B \), and \( C \) have lengths of:

- \( A = 4.51 \, \text{km} \)
- \( B = 2.47 \, \text{km} \)
- \( C = 3.12 \, \text{km} \)

The vector sum of all sides is equal to zero, meaning:

\[
\vec{A} + \vec{B} + \vec{C} + \vec{D} = 0
\]

This means that vector \( D \) will close the quadrilateral.

### Breaking vectors into components:
Using the diagram, we can decompose the vectors into \( x \)- and \( y \)-components based on the angles provided:

1. **Vector A**:
   - Angle = 16° from the positive x-axis.
   - \( A_x = 4.51 \cos(16^\circ) \)
   - \( A_y = 4.51 \sin(16^\circ) \)

2. **Vector B**:
   - Angle = 7.5° below the positive x-axis (subtract from 360° for negative y-component).
   - \( B_x = 2.47 \cos(7.5^\circ) \)
   - \( B_y = -2.47 \sin(7.5^\circ) \)

3. **Vector C**:
   - Angle = 19° above the negative x-axis (180° - 19° for the calculation).
   - \( C_x = -3.12 \cos(19^\circ) \)
   - \( C_y = 3.12 \sin(19^\circ) \)

Once we calculate the components, we can find vector \( D \)'s components using the fact that the sum of all components in both x- and y-directions must be zero:

\[
D_x = -(A_x + B_x + C_x)
\]
\[
D_y = -(A_y + B_y + C_y)
\]

Finally, the magnitude of vector \( D \) can be calculated using the Pythagorean theorem:

\[
D = \sqrt{D_x^2 + D_y^2}
\]

Let's calculate this step by step. I will proceed with the calculations now.The length of the vector \( D \) is approximately \( 4.30 \, \text{km} \).

Would you like more details on the calculations, or do you have any other questions?

Here are 5 related questions you might find helpful:

1. How do we calculate the orientation (angle) of vector \( D \)?
2. What is the significance of vector components in physics and geometry?
3. How would the result change if the angles were altered slightly?
4. Why do we use trigonometric functions to decompose vectors into components?
5. Can you explain how the Pythagorean theorem helps in finding the magnitude of a vector?

**Tip:** Always remember to convert angles to radians when using trigonometric functions in most programming languages, as they typically expect radian input.

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, labeled as A = 4.51 km, B = 2.47 km, and C = 3.12 km, and needs to calculate the length of the fourth side D.

This problem demonstrates how to calculate the length of the fourth side (vector D) of a four-sided plot of land using vector addition, trigonometric decomposition, and the Pythagorean theorem. Given lengths of sides A = 4.51 km, B = 2.47 km, and C = 3.12 km, you will learn how to decompose vectors into components and compute the magnitude of the remaining vector D.

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