A small plane takes off from island A and flies in a straight line for 12 kilometers. At the same time, a sailor sitting in a sailboat who is 5 miles from the island measures the angled by from island A to the sailboat and the plane is 37 degrees. How far is the plane from island B? Please draw and label the situation if possible.
The wording is quite muddled in the middle of your post.
However, I think we have a cosine law problem here, with sides 12 and 5 and the contained angle as 37 degrees
d = 12^2 + 5^2 - 2(12)(5)cos37
= ....
To solve this question, we can use trigonometry, specifically the cosine function. The given information tells us that the angle between the line connecting the plane and the sailboat and the line connecting the plane and island A is 37 degrees. We can use this information to find the distance from the plane to island B.
Let's start by drawing a diagram to better visualize the situation:
Plane
/------------------------------------
/ 37° \
/ \
/ \
/ \
Sailboat Island A
In this diagram, the plane, sailboat, and island A are all in a straight line. The angle marked with 37° is the angle of interest.
Now, let's solve the problem step by step:
Step 1: Convert 5 miles to kilometers.
Since the distance given for the sailboat is in miles, we need to convert it to kilometers since the distance given for the plane is in kilometers.
1 mile is approximately equal to 1.609 kilometers.
Therefore, 5 miles is equal to 5 * 1.609 = 8.045 kilometers.
Step 2: Use trigonometry to find the distance from the plane to island B.
In this case, we can use the cosine function:
cos(angle) = adjacent side / hypotenuse
We know the adjacent side is 8.045 kilometers and the angle is 37 degrees. We want to find the hypotenuse, which represents the distance from the plane to island B.
cos(37°) = 8.045 / hypotenuse
Rearranging the equation, we get:
hypotenuse = 8.045 / cos(37°)
Using a calculator, we can calculate the value of the hypotenuse:
hypotenuse ≈ 10.125 kilometers
Therefore, the plane is approximately 10.125 kilometers away from island B.
Note: In this explanation, I assumed you meant miles when you used the term "miles" in the question. However, if you meant nautical miles, the conversion factor from nautical miles to kilometers is different. Let me know if you need the calculation for nautical miles instead.