1.82 A fence post is 52.0 m from where you are standing, in a direction 37.0 degrees north of east. A second fence post is due south from you. What is the distance of the second post from you, if the distance between the two posts is 80.0 m?

1.82 A fence post is 52.0 m from where you are standing, in a direction $37.0^o$ north of east. A second fence post is due south from you. What is the distance of the second post from you, if the distance between the two posts is 80.0 m?

SOLUTION

Let the point of origin be from where you are standing, $\vec{A}=52.0 \ m$, $37.0^o$ north of east, $\vec{B}$ be the vector to the second post and $\vec{C}$ has a magnitude of $80.0 \ m$. Refer to Figure 1 for a rough sketch of the situation. In this figure,

Let $\vec{d}=80.0 \ m=\sqrt{(d_x)^2+(d_y)^2}$, then its components

$d_x=A_x+B_x=(52.0 \ m) \ cos \ 37.0^o+0=41.529 \ m$

$d_y=A_y+B_y=(52.0 \ m) \ sin \ 37.0^o+B_y=31.294 \ m+B_y$

Then $B_y$ is

$(80.0 \ m)^2=(d_x)^2+(d_y)^2$

$\sqrt{(80.0 \ m)^2-(41.529 \ m)^2}=(31.294 \ m+B_y)$

$B_y=-31.294 \ m+\sqrt{(80.0 \ m)^2-(41.529 \ m)^2}$

$B_y=37.1 \ m$

The length of $\vec{B}$ is $37.1 \ m$. You can check your answer using the law of cosines.