The captain of a freighter 8 km from the nearer of two unlaoding docks on the shore finds that the angle between the lines of sight to the two docks is 35 degrees. if the docks are 10km apart, how far is the tanker from the farther dock?
You can use the law of sines here, on a triangle formed by the ship and the two docks.
(sin 35)/10 = sin B/8
Use that to solve for sin B, which is the angle opposite the 8 km side of the triangle. Then solve for third angle C using
C = 180 - 35 - B
Finally, use
sin A/10 = sin C/c
c is the side length that you want.
Let's assume that the captain of the freighter is at point A, the nearer dock is at point B, and the farther dock is at point C.
We know that AB = 8 km, BC = 10 km, and we're looking for the distance between the freighter (A) and the farther dock (C).
Let's use the Law of Cosines to solve for AC:
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(angle BAC)
The angle BAC is the angle between the lines of sight to the two docks, which is given as 35 degrees.
AC^2 = 8^2 + 10^2 - 2 * 8 * 10 * cos(35)
AC^2 = 64 + 100 - 160 * cos(35)
AC^2 ≈ 36.599
Taking the square root of both sides:
AC ≈ √36.599
AC ≈ 6.05 km
Therefore, the distance between the freighter and the farther dock is approximately 6.05 km.
To solve this problem, we can use the concept of trigonometry and specifically, the Law of Cosines.
Let's define the distances as follows:
- Let d1 be the distance from the captain to the nearer dock.
- Let d2 be the distance from the captain to the farther dock.
- Let x be the distance between the nearer dock and the captain's position.
From the problem, we are given:
- The distance between the two docks is 10 km (d2 - d1 = 10 km).
- The angle between the lines of sight to the two docks is 35 degrees.
We need to find the value of d2, the distance from the captain to the farther dock.
Using the Law of Cosines, we can write the equation:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, our sides and angles are as follows:
- Side a (c): 10 km (distance between the two docks)
- Side b (x): d1 km (the distance from captain to the nearer dock)
- Angle C: 35 degrees
Applying the Law of Cosines:
d2^2 = x^2 + 10^2 - 2 * 10 * x * cos(35)
Now we know that x = d1, so we can substitute it:
d2^2 = d1^2 + 100 - 20 * d1 * cos(35)
Since we are given that x = 8 km, we can substitute d1 with 8:
d2^2 = 8^2 + 100 - 20 * 8 * cos(35)
Simplifying further:
d2^2 = 64 + 100 - 160 * cos(35)
d2^2 = 164 - 160 * cos(35)
Now, we can calculate the value of d2 by taking the square root:
d2 = sqrt(164 - 160 * cos(35))
By evaluating this equation, we find that d2 ≈ 4.42 km. Therefore, the distance from the captain to the farther dock is approximately 4.42 km.