Answer
55.5 m
Work Step by Step
First, we convert minutes into decimal degrees:
$51^{\circ}20'=51\frac{20}{60}=51.33^{\circ}$
We can use the law of cosines here because we know the lengths of two sides of the triangle and the measure of the included angle.
The law of cosines is:
$a^{2}=b^{2}+c^{2}-2bc\cos A$
where $b,c$ are the two known sides of the triangle while $A$ is the known angle. The unknown side opposite the known angle is $a$.
Substituting the values in the formula and solving:
$a^{2}=b^{2}+c^{2}-2bc\cos A$
$a^{2}=58.2^{2}+68.3^{2}-2(58.2)(68.3)\cos 51.33$
$a^{2}=3387.24+4664.89-7950.12\cos 51.33$
$a^{2}=8052.13-7950.12\cos 51.33$
Using a calculator, we find that $\cos 51.33^{\circ}=0.62483$. Therefore,
$a^{2}=8052.13-7950.12\cos 51.33$
$a^{2}=8052.13-7950.12(0.62483)$
$a^{2}=8052.13-4967.50$
$a^{2}=3084.63$
$a=\sqrt {3084.63}$
$a=55.54\approx55.5$
Therefore, the length of the unknown side of the triangle is 55.5 m.
