Answer
The length of the wire on the upper side of the hill is 28.6 feet
The length of the wire on the lower side of the hill is 37.9 feet
Work Step by Step
Let $c$ be the length of the wire on the upper side of the hill.
Let $a = 15~ft$
Let $b = 30~ft$
Let $C$ be the angle between these two sides. Then $C = 90^{\circ}-20^{\circ}= 70^{\circ}$
We can use the law of cosines to find $c$:
$c^2 = a^2+b^2-2ab~cos~C$
$c = \sqrt{a^2+b^2-2ab~cos~C}$
$c = \sqrt{(15~ft)^2+(30~ft)^2-2(15~ft)(30~ft)~cos~70^{\circ}}$
$c = 817.18~ft^2$
$c = 28.6~ft$
The length of the wire on the upper side of the hill is 28.6 feet
Let $d$ be the length of the wire on the lower side of the hill.
Let $a = 15~ft$
Let $b = 30~ft$
Let $D$ be the angle between these two sides. Then $D = 90^{\circ}+20^{\circ}= 110^{\circ}$
We can use the law of cosines to find $d$:
$d^2 = a^2+b^2-2ab~cos~D$
$d = \sqrt{a^2+b^2-2ab~cos~D}$
$d = \sqrt{(15~ft)^2+(30~ft)^2-2(15~ft)(30~ft)~cos~110^{\circ}}$
$d = 1432.82~ft^2$
$d = 37.9~ft$
The length of the wire on the lower side of the hill is 37.9 feet
