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.Thanks!
To solve this problem, we can use trigonometry. Let's denote the distance from the plane to island B as x kilometers.
To visualize the situation, let's draw a diagram.
A—————————x—————————B
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plane sailboat
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A———————————12 km———————————
In this diagram:
- The distance from island A to island B is labeled as 12 km.
- The distance from the sailboat to island B is labeled as 5 miles, but we'll need to convert it to kilometers later.
- The angle between the plane and the line connecting the sailboat to island B is 37 degrees.
To find the distance from the plane to island B (x), we can use the sine function:
sin(37 degrees) = distance from the sailboat to island B / distance from the plane to island B
Let's solve for x:
sin(37 degrees) = 5 miles / x
sin(37 degrees) = 5 km / x (since we need to use kilometers)
To solve for x, we can rearrange the equation:
x * sin(37 degrees) = 5 km
Now, let's substitute the values and calculate x:
x = 5 km / sin(37 degrees)
x ≈ 8.24 km
Hence, the plane is approximately 8.24 kilometers away from island B.
To solve this problem, we can use trigonometry, specifically the sine function. Let's label the diagram to better visualize the situation:
A B
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|------d1------|------d2------|---12 km---| |
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|_______37°________|_________|________|
In this diagram, A represents the initial position of the plane, B represents the final position of the plane, and d1 represents the distance between the plane and the sailboat.
Now, since we have a right-angled triangle, we can use the sine function to find the length of d1. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse.
So, in this case, sin(37°) = d1 / 5 miles
To find the value of d1, we rearrange the equation:
d1 = sin(37°) * 5 miles
Using a calculator, we can find the sine of 37° and then multiply it by 5 miles to get the distance d1.
Finally, to find the distance from island B to the plane, we need to subtract the distance traveled by the plane (12 kilometers) from the distance between the plane and the sailboat (d1):
Distance from island B = d1 - 12 kilometers
Again, using a calculator, we can subtract the two distances to find the final answer.