A ship is tied to two anchors on the shoreline. The anchor line on the left forms a 46 degree angle with the shoreline and is 350 feet long. The anchor line on the right forms a 63 degree angle with theshoreline and is 283 feet long. Find the exact distance (in feet) between the two anchors.
Please show all work
use the Law of Sines
the angle at the ship is
... 180º - 46º - 63º
283 ft / sin(43º) = 350 ft / sin(63º)
... = x / sin(180º - 46º - 63º)
i mis-transposed the 43º
... it should be 46º
To find the exact distance between the two anchors, we can use the Law of Cosines, which states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice their product, multiplied by the cosine of the included angle.
Let's denote the distance between the two anchors as 'd'.
Applying the Law of Cosines to the triangle formed by the left anchor line, the right anchor line, and the distance 'd', we have:
d^2 = 350^2 + 283^2 - 2(350)(283)cos(180° - 46° - 63°)
First, let's simplify the cosine term:
cos(180° - 46° - 63°) = cos(71°)
Now, let's substitute the values into the equation and calculate:
d^2 = 122500 + 79989 - 2(350)(283)cos(71°)
d^2 = 122500 + 79989 - 197800cos(71°)
d^2 = 202489 - 197800(cos(71°))
Next, we need to calculate the cosine of 71°:
cos(71°) = 0.32705 (rounded to five decimal places)
Substituting this value into the equation:
d^2 = 202489 - 197800(0.32705)
d^2 = 202489 - 64759.51
d^2 = 137729.49
Finally, we take the square root of both sides to find the exact distance 'd':
d = √137729.49
d ≈ 371.28 feet
Therefore, the exact distance between the two anchors is approximately 371.28 feet.