An ocean liner is 177 miles due west of lighthouse A. Lighthouse B is 265 miles due south of lighthouse A. Find the distance from lighthouse B to the liner and the bearing of the ocean liner from lighthouse B.
distance is √(177^2 + 265^2)
bearing is (90-θ) where tanθ = -265/177
To solve this problem, we can use the Pythagorean theorem to find the distance between Lighthouse B and the ocean liner.
Let's consider the triangle formed by Lighthouse A, Lighthouse B, and the ocean liner:
Using the Pythagorean theorem, we have:
Distance^2 = (Distance from Lighthouse A to the liner)^2 + (Distance from Lighthouse B to the liner)^2
Let's calculate the distance from Lighthouse A to the liner:
Distance from Lighthouse A to the liner = 177 miles (given)
Next, let's calculate the distance from Lighthouse B to the liner:
Distance from Lighthouse B to the liner = 265 miles (given)
Now we can substitute these values into the Pythagorean theorem to find the distance between Lighthouse B and the liner:
Distance^2 = 177^2 + 265^2
Calculating:
Distance^2 = 31329 + 70225
Distance^2 = 101554
Taking the square root of both sides, we get:
Distance ≈ 318.95 miles
So, the distance from Lighthouse B to the ocean liner is approximately 318.95 miles.
Now let's calculate the bearing of the ocean liner from Lighthouse B.
To find the bearing, we need to use trigonometry, specifically the tangent function (tan).
The bearing is the angle formed between the line connecting Lighthouse B and the ocean liner, and the line pointing due north (0 degrees).
Using the tangent function:
Tan(Bearing) = (Distance from Lighthouse A to the liner) / (Distance from Lighthouse B to the liner)
Substituting the given values:
Tan(Bearing) = 177 / 265
Calculating:
Bearing ≈ 35.3 degrees
So, the bearing of the ocean liner from Lighthouse B is approximately 35.3 degrees.
To find the distance from lighthouse B to the liner, we can use the Pythagorean theorem since we have a right triangle formed by the two lighthouses and the ocean liner.
Let's call the distance from lighthouse B to the ocean liner "x" (in miles).
Using Pythagorean theorem:
x^2 = (177 miles)^2 + (265 miles)^2
Simplifying the equation:
x^2 = 31329 miles^2 + 70225 miles^2
x^2 = 10155454 miles^2
Taking the square root of both sides to find x:
x = √(10155454 miles^2)
x ≈ 3188.52 miles
Therefore, the distance from lighthouse B to the ocean liner is approximately 3188.52 miles.
To find the bearing of the ocean liner from lighthouse B, we can use trigonometry.
Let's call the angle between the line connecting lighthouse B and the ocean liner and the line connecting lighthouse B and lighthouse A "θ" (in degrees).
Using trigonometry:
tan(θ) = opposite / adjacent
tan(θ) = 177 miles / 265 miles
tan(θ) ≈ 0.669811
Now we can find the value of θ by taking the inverse tangent (arctan) of both sides:
θ ≈ arctan(0.669811)
Using a calculator, we find that:
θ ≈ 33.75 degrees
Therefore, the bearing of the ocean liner from lighthouse B is approximately 33.75 degrees.