Question

10: 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 Figure 16, and then correctly calculates the length and orientation of the fourth side D What is his result? 3.02 km 19 D- km 16° 7.5 B-2.48 km A - 4.70 km Figure 16.

Answer 1

We know that, $\vec{A}$ + $\vec{B}$ + $\vec{C}$ + $\vec{D}$ = 0

$\vec{D}$ = - $\vec{A}$ - $\vec{B}$ - $\vec{C}$

The x-component of the length of fourth side $\vec{D}$ which will be given as -

$\vec{D}$_x = - $\vec{A}$_x - $\vec{B}$_x - $\vec{C}$_x

$\vec{D}$_x = - (A cos 7.5⁰) - (-B sin 16⁰) - (-C cos 19⁰)

$\vec{D}$_x = - (4.70 km) (0.9914) + (2.48 km) (0.2756) + (3.02 km) (0.9455)

$\vec{D}$_x = - (4.65958 km) + (0.683488 km) + (2.85541 km)

$\vec{D}$_x = - 1.120682 km

The y-component of the length of fourth side $\vec{D}$ which will be given as -

$\vec{D}$_y = - $\vec{A}$_y - $\vec{B}$_y - $\vec{C}$_y

$\vec{D}$_y = - (-A sin 7.5⁰) - (B cos 16⁰) - (C sin 19⁰)

$\vec{D}$_y = - (-4.70 km) (0.1305) - (2.48 km) (0.9613) - (3.02 km) (0.3256)

$\vec{D}$_y = (0.61335 km) - (2.384024 km) - (0.983312 km)

$\vec{D}$_y = - 2.753986 km

Therefore, magnitude of the length of fourth side $\vec{D}$ which will be given as -

| $\vec{D}$ | = _$\sqrt{}$$\vec{D}$_x² + $\vec{D}$_y²

| $\vec{D}$ | = _$\sqrt{}$(-1.120682 km)² + (-2.753986 km)²

| $\vec{D}$ | = 2.97 km

And orientation of the fourth side $\vec{D}$ which will be given as -

$\theta$ = tan^-1 ($\vec{D}$_x / $\vec{D}$_y)

$\theta$ = tan^-1 [(-1.120682 km) / (-2.753986 km)]

$\theta$ = tan^-1 (0.4069)

$\theta$ = 22.1 degree