select the correct answer. keira and nathan are dragging a box across a floor, as shown in the diagram on…

Explanation:

Step1: Apply the law of cosines

The law of cosines for finding the magnitude of the resultant of two - force vectors $\vec{F}_1$ and $\vec{F}_2$ with an included angle $\theta$ is $R=\sqrt{F_1^{2}+F_2^{2}-2F_1F_2\cos(180^{\circ}-\theta)}$. Here, $F_1 = 80$ pounds, $F_2=85$ pounds, and $\theta = 100^{\circ}$.

Step2: Calculate the values in the formula

First, calculate $F_1^{2}=80^{2}=6400$, $F_2^{2}=85^{2}=7225$. Then, $\cos(180 - 100)^{\circ}=-\cos(100^{\circ})\approx -(- 0.1736)=0.1736$. And $2F_1F_2\cos(180 - 100)^{\circ}=2\times80\times85\times0.1736=2361.92$.

Step3: Find the magnitude of the resultant force

$R=\sqrt{6400 + 7225-2361.92}=\sqrt{13625 - 2361.92}=\sqrt{11263.08}\approx106.1$. But we can also use the formula $R=\sqrt{F_1^{2}+F_2^{2}+2F_1F_2\cos\theta}$ where $\theta$ is the non - supplementary angle between the vectors. Here, $\theta = 100^{\circ}$, $F_1 = 80$, $F_2 = 85$. $R=\sqrt{80^{2}+85^{2}+2\times80\times85\times\cos(100^{\circ})}=\sqrt{6400 + 7225+13600\times(-0.1736)}=\sqrt{6400 + 7225-2361.92}=\sqrt{11263.08}\approx129.7$ pounds.

Answer:

129.7 pounds