[Math] Trigonometric problem involving 2 ships and bearing and distance

trigonometry

Hello im doing this problem ships $a$ and $b$, $100$ miles apart are traveling 60mph and 80 mph, respectively. At 12NN, the bearing of $A$ from B is $72^{\circ}$. Find the distance and bearing of B from A at 5:30PM. I understand what i need to do and i just need help if my illustration is correct here is my illustration
My illustration link here

Thanks in advance

Best Answer

Here is the general way to solve these questions:

Since you know the speed of each boat and the time that they have travelled since being however far they are from each other (In this case, 100 miles from each other at 72 degrees), then you can determine how far they would have travelled given a specified direction. You can then compare the x and y components in order to determine how far they are from each other. Hope this helps.

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[Math] Trigonometry Airplane question. Finding bearing and distance.

Cosine Law would be easier, but let's do it your way. Consider the small right triangle (call it $\Delta ABC$), which roughly looks like:

A----B
|   /
|  /
| /
|/
C

Note that $\angle ABC =59^\circ$. Thus, $AB=640\cos(59^\circ)$ and $AC=640\sin(59^\circ)$.

Now consider the large right triangle. Its two legs are $AC$ and $AB+480$. Hence, by the Pythagorean Theorem, we have: $$x=\sqrt{(AC)^2 + (AB+480)^2}=\sqrt{(640\cos(59^\circ)+480)^2 + (640\sin(59^\circ))^2} \approx 978 \text{ miles}$$ Finally, for the angle bearing: $$\theta=\tan^{-1}\left(\frac{AC}{AB+480}\right) = \tan^{-1}\left(\frac{640\sin(59^\circ)}{640\cos(59^\circ)+480}\right) \approx 34^\circ \text{ to get the bearing you subtract that value from 90 to get } S56^\circ W$$

[Math] Bearings Problem

Won't you calculate the degrees from the east direction?. If you did, it is simple application of pythogorus theorem. It travels 2*35 nm at 53 degress to the east. Then it travels 3*35 nm at 143 degrees to the east. It's position is $\sqrt{70^2 + 105^2} = 126.2$ miles from the port of cleaveland and it is 109.3 degrees to the east.

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Related Solutions

[Math] Trigonometry Airplane question. Finding bearing and distance.

[Math] Bearings Problem

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