a boat leaves lighthouse P and sails 10 miles. At the same time, it is sighted from lighthouse Q, 13 miles west of P. The bearing of the ship from Q is North 70 degree and 30 minutes East. Find the distance of the ship from Q.

sin70.5 = 13/d

d = 13/sin70.5 = 13.8 Miles.

To find the distance of the ship from lighthouse Q, we can use the Law of Sines. Let's break down the given information:

Side a = 10 miles (distance from lighthouse P)
Side b = distance of the ship from Q (what we want to find)
Angle A = North 70 degrees 30 minutes East (bearing from Q)

First, we need to convert the angle from degrees and minutes to decimal degrees.

70 degrees + 30 minutes = 70 + 30/60 = 70.5 degrees

Now, using the Law of Sines:

Sin(A)/a = Sin(B)/b

We can plug in the known values:

Sin(70.5°)/10 = Sin(90°)/b

Solving for b:

b = (10 * Sin(90°))/Sin(70.5°)

Using a calculator:

b ≈ (10 * 1)/0.9511

b ≈ 10.512 miles

Therefore, the distance of the ship from lighthouse Q is approximately 10.512 miles.

To find the distance of the ship from lighthouse Q, we can use trigonometry and construct a triangle with lighthouse Q, the ship, and the point where it was sighted. Let's call this point S.

First, let's find the angle between lighthouses P and Q. Since the ship is sighted from lighthouse Q, which is 13 miles west of P, and it sailed a distance of 10 miles, we can use the inverse tangent function (tan^(-1)) to find the angle.

tan^(-1)(10/13) ≈ 38.66 degrees

So, the angle between lighthouses P and Q is approximately 38.66 degrees.

Now, let's find the bearing of the ship from lighthouse P by subtracting the angle between lighthouses P and Q, 38.66 degrees, from the bearing of the ship from Q, which is North 70 degrees and 30 minutes East.

North 70 degrees and 30 minutes East - 38.66 degrees ≈ North 31 degrees and 30 minutes East

Therefore, the bearing of the ship from lighthouse P is North 31 degrees and 30 minutes East.

Now, we can use the Law of Cosines to find the distance of the ship from lighthouse Q.

In triangle QPS:
QS^2 = PQ^2 + PS^2 - 2(PQ)(PS)cos(PQS)

Since PQ (the distance between lighthouses P and Q) is 13 miles, PS (the distance traveled by the ship) is 10 miles, and PQS (the angle between PQ and PS) is 31 degrees and 30 minutes, we can plug these values into the equation.

QS^2 = 13^2 + 10^2 - 2(13)(10)cos(31°30')
QS^2 = 169 + 100 - 260cos(31.5°)
QS^2 ≈ 169 + 100 - 260(0.867)
QS^2 ≈ 169 + 100 - 225.22
QS^2 ≈ 43.78

Taking the square root of both sides, we can find the distance of the ship from lighthouse Q.

QS ≈ √43.78
QS ≈ 6.62 miles

Therefore, the distance of the ship from lighthouse Q is approximately 6.62 miles.