A naval officer sighted the smoke of a volcanic island on a bearing of 044 degrees true. A navigator on another ship 25 km due east of the first ship saw the smoke on a bearing of 342 degrees true. Find the distance of each ship from the volcano.

Question

anonymous · Answer

|dw:1376910460581:dw| In the image you see a rough outline of your problem. Effectively you have a tri-angle with a 25 km base between the ships. The directions to the island give you two of the three angles within the triangle"

Angle at Ship1 : 90 deg - 44 deg = 46 deg
Angle at Ship2 : 342 deg - 270 deg = 72 deg (do you see why ?)

With two angles you can calculate the third angle (180 - angle1 - angle2). If length of the sides is known, you can calculate the length of the other sides. These are typically the distances from both ships to the Island. Let me know if you get stuck somewhere.

anonymous · Answer

I do understand infact I have the exact same sketch on my page however when i plug the values into the sin rule I still manage to get the incorrect answers. The answers are 19.89km and 18.16, btw.

anonymous · Answer

let me check quickly, I didn't do the calculations yet.

anonymous · Answer

Wasn't sure if you were familiar with the sine rule. That's the one you need to apply. My top angle is 180 deg - (4+72) = 62 deg, correct ? If so, you can use the known angle and know leg length (62 degrees go with the 25 km) to calculate the leg from Ship1 to the Island as follows:

Dship1 / sin(72) = 25 / sin(62), but you knew that already :-). This means that the distance from Ship1 to the Island is 25 * (sin(72)/sin(62)) = 25 * 1.077 = 26.9 km. Is that what you got too ?

anonymous · Answer

Sorry, mistake in the first section above. The top angle is 180-(46+72)=62 deg

anonymous · Answer

yes actually i did get 26.9 maybe its just a printing error in the book, i mean its a cambridge book so its unlikely, but what are the chances that we both got the same answer and both happen to be wrong

anonymous · Answer

And how about the second distance ? Calculated in the same way: 20.36 km. You never know about printing errors, but I agree with you that we both arrived at a solution in the same way. I believe the method is correct, therefore. Do you have any other examples with answers that we can use to verify ?

anonymous · Answer

yes i have both those answers. Unfortunately no, there is no other way to verify ill just have to run this question by my teacher

anonymous · Answer

Thanks for your help though its really appreciated

anonymous · Answer

You're welcome, good luck :-)