## Farmer wants to Fence off his Four-Sided Plot with missing Side| Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.22

### A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in the figure, and then correctly calculates the length and orientation of the fourth side D. What is his result? ## A Landowner with Triangular Piece of Flat Land| Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.20

### A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors A from B in Figure 3.61. She then correctly calculates the length and orientation of the third side C. What is her result? ## Solution:

### Part A

Consider the following figure:

The east distance is the component in the horizontal direction. $D_E=7.50\:km\:\cdot sin\:\left(15^{\circ} \right)$ $D_E=1.94\:km$

The north distance is the vertical component $D_E=7.50\:km\cdot cos\left(15^{\circ} \right)$ $D_E=7.24\:km$

### Part B

Based from the figure, we can easily see that the order is reversible in the addition of vectors. We say that $D_E+D_N=D_N+D_E$

## Solution:

Consider the right triangle formed by the legs A, B, and R. We know that A is 18 m, B is 25 m, and we are solving for the magnitude of R. We can do this by using the Pythagorean Theorem. That is $R=\sqrt{A^2+B^2}$ $R=\sqrt{\left(18\:m\right)^2+\left(25\:m\right)^2}$ $R=\sqrt{324+625}$ $R=\sqrt{949}$ $R=30.8\:m$

So, the distance is about 30.8 meters from the starting point. To solve for the value of the unknown angle, φ, we can use the tangent function. That is $tan\:\phi =\frac{A}{B}$ $tan\:\phi =\frac{18\:m}{25\:m}$ $\phi =tan^{-1}\left(\frac{18}{25}\right)$ $\phi =35.75^{\circ}$

Therefore, the compass angle is $35.75^{\circ} \:West\:of\:North$