Question

A spelunker is surveying a cave. She follows a passage 180 m straight west, then 210 m in a direction 450 east of south, and then 280 m at 300 east of north. After a fourth unmeasured displacement, she finds herself back where she started. Use a scale drawing to determine the magnitude and direction of the fourth displacement.

Answer 1

Let's draw all the spelunker's displacements and find the unmeasured displacement:

Let's find the unknown $x$ and $y$ components of the unmeasured displacement of the spelunker:

$$d_x=180\ m\cdot cos180^{\circ}+210\ m\cdot cos(360^{\circ}-45^{\circ})+280\ m\cdot cos30^{\circ}+x=0,$$ $$d_y=180\ m\cdot sin180^{\circ}+210\ m\cdot sin(360^{\circ}-45^{\circ})+280\ m\cdot sin30^{\circ}+y=0,$$ $$x=-211\ m, y=8.5\ m.$$

Then, the magnitude of the unmeasured displacement can be found from the Pythagorean theorem:

$$d=\sqrt{d_x^2+d_y^2}=\sqrt{(-211\ m)^2+(8.5\ m)^2}=211.2\ m.$$

We can find the angle as follows:

$$\theta=sin^{-1}(\dfrac{d_y}{d})=sin^{-1}(\dfrac{8.5\ m}{211.2\ m})=2.3^{\circ}.$$

The sign plus means that the unmeasured displacement has direction $2.3^{\circ} N\ of\ W$.

Therefore, the unmeasured fourth displacement $d$ has magnitude 211.2 m and direction of $2.3^{\circ} N\ of\ W$.