A ladder resting against a wall forms an angle of 60 degree with the level ground. If the foot of the ladder is moved out from the wall 8ft. the ago forms with the ground is 45 degree. Find the length of the ladder.

If the foot of the ladder started out at x ft away from the wall, then we know that the length of the ladder is 2x ft.

With a 45° angle, then the two legs are x+8 ft, and the length of the ladder is (x+8)√2.

So, now we have

2x = (x+8)√2
x(2-√2) = 8√2
x = 8√2/(2-√2)

the ladder is thus

16√2/(2-√2) = 16(1+√2) feet long

To find the length of the ladder, we can apply some basic trigonometry concepts.

Let's assume the length of the ladder is represented by 'x'.

When the ladder is resting against the wall and forms an angle of 60 degrees with the level ground, we can use the sine function:

sin(60°) = opposite / hypotenuse.

In this case, the opposite side is the height of the wall, and the hypotenuse is the length of the ladder. So we have:

sin(60°) = height / x.

Now, let's move to the second scenario where the foot of the ladder is moved out from the wall by 8ft, and the angle formed with the ground is 45 degrees. Again, we can use the sine function:

sin(45°) = opposite / hypotenuse.

In this case, the opposite side is the height of the wall (which remains the same), but the hypotenuse is now (x + 8) ft. So we have:

sin(45°) = height / (x + 8).

We need to find the value of 'x' that satisfies both equations. We can start by solving the first equation for the height:

height = x * sin(60°).

Substitute this expression for height into the second equation:

sin(45°) = x * sin(60°) / (x + 8).

Now we can solve this equation to find the value of 'x'. First, multiply both sides by (x + 8):

sin(45°) * (x + 8) = x * sin(60°).

Expand the left side:

(sin(45°) * x) + (sin(45°) * 8) = x * sin(60°).

Multiply both sides by sin(60°):

sin(60°) * sin(45°) * x + sin(60°) * sin(45°) * 8 = x * sin(60°) * sin(45°).

Simplify:

x * (sin(60°) * sin(45°) - sin(60°) * sin(45°)) = -sin(60°) * sin(45°) * 8.

Divide both sides by (sin(60°) * sin(45°)):

x = (-sin(60°) * sin(45°) * 8) / (sin(60°) * sin(45°)).

After substituting the values of the trigonometric functions, we can simplify the equation to solve for 'x'. Ultimately, x will represent the length of the ladder.