from the top of a cliff 250 ft. high the angle of depression of a boat is 10 degrees. How far out is the boat from the foot of the cliff?

1417.82 feet

2.9

From the top of cliff 250 feet high the angle of depression of a boat is 10° how far out is the boat to the cliff

Well, well, well, looks like we have a high flyer over here! So, let's calculate this. Since we know the angle of depression is 10 degrees, and the height of the cliff is 250 feet, we can imagine a right triangle with the cliff height as the vertical side. Now we just need a good ol' trigonometric helper, the tangent function!

Using some trigonometry magic, we can say that tan(10 degrees) = opposite/adjacent, where the opposite side is the height of the cliff and the adjacent side is the distance from the foot of the cliff to the boat (let's call it x).

So, we have tan(10 degrees) = 250 / x. Now, we can solve this equation to find x.

But hey, don't fret! Let me calculate it for you. Give me a moment...

Calculating... crunching numbers... hilarious calculation noises...

Alright! According to my calculations, the boat is approximately 1433.53 feet away from the foot of the cliff.

Now that's what I call a boat with some distance!

To find the distance between the boat and the foot of the cliff, we need to use trigonometry and specifically the tangent function.

Let's denote the distance between the boat and the foot of the cliff as "x" (in feet). We have an angle of depression of 10 degrees, which means that the angle between the horizontal line and the line of sight from the top of the cliff to the boat is 10 degrees.

Now, we can set up a right triangle with the vertical distance (height of the cliff) as the opposite side and the horizontal distance (between the boat and the foot of the cliff) as the adjacent side, with the angle of depression being the angle opposite the vertical distance.

Using the tangent function:

tan(angle) = opposite / adjacent

In this case:

tan(10 degrees) = 250 ft / x

Now, we can solve for x by rearranging the equation:

x = 250 ft / tan(10 degrees)

Calculating this using a scientific calculator, we get:

x ≈ 1432.79 ft

Therefore, the boat is approximately 1432.79 feet away from the foot of the cliff.

1439.6926 ft

tan 10° = 250/x

x = 250/tan10 = .....