Lighthouse B is 8 miles west of lighthouse A. A boat leaves A and sails 5 miles. At this time, it is sighted from B. If the bearing of the boat from B is N63E, how far from B is the boat?
The boat is either ___ miles or ____ miles from lighthouse B, to the nearest tenth of a mile.
11,3.7
To find the distance from the boat to lighthouse B, we can use trigonometry and the given information.
We know that the boat is sighted at a bearing of N63E from lighthouse B. This means that the angle between the line connecting lighthouse B to the boat and the north direction is 63 degrees.
Since the boat is 8 miles west of lighthouse A and sails 5 miles, it must be 3 miles west of its original position (8 miles - 5 miles).
To find the distance from the boat to lighthouse B, we can create a right triangle with the boat on one side, the line connecting lighthouse B to the boat as the hypotenuse, and a vertical line from the boat to the bearings line as the other side.
Using trigonometry, we can use the sine function to find the length of the opposite side (x) from the given angle (63 degrees) and the known length of the adjacent side (3 miles).
sin(63 degrees) = opposite / hypotenuse
sin(63 degrees) = x / 3
Rearranging the equation:
x = sin(63 degrees) * 3
Calculating the value:
x ≈ 2.70 miles
Therefore, to the nearest tenth of a mile, the boat is either 2.7 miles or 2.7 miles from lighthouse B.
To solve this problem, we can use the concept of bearings and the Law of Sines. Let's break down the steps:
1. Draw a diagram: Draw a diagram with lighthouse A and lighthouse B located horizontally, and mark the position of the boat after sailing 5 miles from A.
2. Determine the bearing angle: The bearing angle N63E can be translated into a triangle in the diagram. The angle between North and the boat's bearing from B is 90 - 63 degrees, which gives us 27 degrees.
3. Calculate the angle at B: Since the triangle is a right-angled triangle, the angle at B is 180 - 90 - 27 degrees, which is 63 degrees.
4. Apply the Law of Sines: We can use the Law of Sines to find the distance from the boat to B. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. In this case, we can set up the following proportion:
(Length of side opposite angle B) / sin(63 degrees) = (Distance from B to boat) / sin(90 degrees)
Rearranging the equation, we get:
Distance from B to boat = (Length of side opposite angle B) * (sin(90 degrees) / sin(63 degrees))
5. Calculate the length of the side opposite angle B: Since the boat sailed 5 miles, the length of the side opposite angle B is 8 - 5 = 3 miles.
6. Calculate the distance from B to the boat: Substituting the values into the equation, we get:
Distance from B to boat = 3 miles * (sin(90 degrees) / sin(63 degrees))
7. Use a calculator to find the sin value for each angle: sin(90 degrees) = 1, and sin(63 degrees) ≈ 0.89.
8. Substitute the sin values and calculate:
Distance from B to boat = 3 miles * (1 / 0.89) ≈ 3.37 miles
Therefore, the boat is approximately 3.37 miles from lighthouse B.