A light house is 10 miles northeast of a dock. A ship leaves a dock at moon, and sails at a speed of 12 miles an hour. At what time will it be 8 miles from the lighthouse?

since the lighthouse is 10 mi NE of the dock, it is 7.07mi east and 7.07 mi north. (legs of a 45-45-90 triangle)

Now, if the ship is 8 mi from the lighthouse, we have a new x-7.07-8 triangle, so x = 3.74.

Now, the distance from the dock to the ship is 7.07 - 3.74 = 3.33 mi.

At 12 mi/hr, that would take 3.33/12 = .277777 hours, or 16'40"

So, at 12:16:40 the ship is 8 mi from the lighthouse

Note: the ship will also be 8 miles from the lighthouse after it has passed it and is east of the lighthouse. From the above data, you can figure the second time with no trouble.

To find out the time when the ship will be 8 miles from the lighthouse, we need to calculate the distance covered by the ship.

We can start by constructing a right triangle, where the dock is at the vertex of the right angle, the lighthouse is at one of the other vertices, and the ship's current position is at the third vertex. We know that the distance between the dock and the lighthouse is 10 miles.

Given that the ship is sailing at a speed of 12 miles per hour, we can use the Pythagorean theorem to calculate the distance it has traveled.

Let's denote the distance from the dock to the ship's current position as x. Using the Pythagorean theorem, we have:

x^2 + 8^2 = 10^2
x^2 + 64 = 100
x^2 = 100 - 64
x^2 = 36
x = √36
x = 6

So, the ship is currently 6 miles away from the dock.

Now, we can calculate the time it will take for the ship to travel the remaining distance of 2 miles (from 6 miles to 8 miles):

Time = Distance / Speed
Time = 2 miles / 12 miles per hour
Time = 1/6 hour = 10 minutes

Therefore, the ship will be 8 miles from the lighthouse in approximately 10 minutes.

To find the time at which the ship will be 8 miles from the lighthouse, we need to first determine the distance the ship needs to cover to reach that point.

We know that the lighthouse is 10 miles northeast of the dock. Assuming the ship leaves from the dock and sails directly towards the lighthouse, it will form a right-angled triangle with the dock, the lighthouse, and its current position.

The distance between the dock and the lighthouse is the hypotenuse of this triangle, while the distance the ship needs to cover is one of the other two sides. We can use the Pythagorean theorem to find this distance.

According to the Pythagorean theorem, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, let's assume the distance the ship needs to cover is side a, and the distance between the lighthouse and the dock is side b.

So we have:
c^2 = a^2 + b^2

Given that side b is 10 miles, we can rearrange the equation to find side a:
a^2 = c^2 - b^2
a^2 = (8 miles)^2 - (10 miles)^2
a^2 = 64 miles^2 - 100 miles^2
a^2 = -36 miles^2

Oh no! It seems that we have encountered an issue. The value of a^2 is negative, which indicates that a^2 does not have a real solution. This means that the ship will not reach a point 8 miles from the lighthouse if it sails directly towards it.

There might be additional information or a different approach to solve this question.