A boat travels for 8 kilometers in a straight line from the dock. It is then sighted from a lighthouse which is 6.5 kilometers from the dock. The angle determined by the dock, the lighthouse(vertex), and the boat is 25 degrees. How far is the boat from the lighthouse?
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To find the distance between the boat and the lighthouse, we can use trigonometry.
Let's call the distance between the boat and the lighthouse, x.
We have a right triangle formed by the boat, the lighthouse, and the straight line from the dock. The angle between the straight line and the line connecting the lighthouse and the boat is 25 degrees.
Using trigonometry, we can use the sine function to relate the angle and the sides of the triangle:
sin(angle) = opposite / hypotenuse
In this case, the opposite side is the distance between the boat and the lighthouse (x), and the hypotenuse is the distance between the dock and the lighthouse (6.5 kilometers).
sin(25 degrees) = x / 6.5
Now, we can solve for x by multiplying both sides by 6.5:
x = 6.5 * sin(25 degrees)
Using a calculator to find sin(25 degrees), we get:
x ≈ 6.5 * 0.4226
x ≈ 2.749 km
Therefore, the boat is approximately 2.749 kilometers away from the lighthouse.
To find the distance between the boat and the lighthouse, we can use the Law of Cosines. The Law of Cosines states that for any triangle with side lengths a, b, and c, and angle C opposite side c, the following formula holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we have a triangle with side lengths 8 kilometers (length from dock to the boat) and 6.5 kilometers (length from dock to the lighthouse), and an angle of 25 degrees (angle opposite side c, the distance between the boat and the lighthouse). We want to find side length c.
Plugging the values into the Law of Cosines formula, we get:
c^2 = 8^2 + 6.5^2 - 2 * 8 * 6.5 * cos(25)
Simplifying further:
c^2 = 64 + 42.25 - 104 * cos(25)
Now, we can calculate the value of c by taking the square root of both sides:
c = sqrt(64 + 42.25 - 104 * cos(25))
Using a calculator, we can evaluate cos(25) and substitute it into the equation:
c ≈ sqrt(64 + 42.25 - 104 * 0.90631)
Simplifying again:
c ≈ sqrt(64 + 42.25 - 94.31624)
c ≈ sqrt(11.93376)
c ≈ 3.45 kilometers
Therefore, the boat is approximately 3.45 kilometers away from the lighthouse.