An extreme skier slides down an incline that forms an angle 24 degrees above the horizontal. The skier starts from rest and accelerates firmly at 4.00m/s^2 for a distance of 50.0m to the edge of a cliff. The cliff is 30.0m above the powdery snow below. Find how far from the bottom of the cliff the skier lands.

To solve this problem, let's break it down into steps:

Step 1: Find the horizontal distance the skier travels before reaching the edge of the cliff.

Given:
- Acceleration (a): 4.00 m/s^2
- Distance (d): 50.0 m

Using the equation of motion: d = v0*t + 0.5*a*t^2, where v0 is the initial velocity and t is the time.

Since the skier starts from rest, the initial velocity (v0) is 0 m/s.

Substituting the given values into the equation, we have:
50.0 m = 0 + 0.5*(4.00 m/s^2)*t^2

Simplifying the equation:
25.0 m = 2.00 m/s^2*t^2

Dividing both sides by 2.00 m/s^2:
t^2 = 12.5 s^2

Taking the square root of both sides:
t = √(12.5) s
t ≈ 3.54 s

Now, let's find the horizontal distance (x) using the equation: x = v0*t + 0.5*a*t^2.

Substituting the values:
x = 0 + 0.5*(4.00 m/s^2)*(3.54 s)^2
x = 0 + 0.5*(4.00 m/s^2)*(12.54 s^2)
x = 0 + 25.08 m
x ≈ 25.1 m

The skier travels approximately 25.1 meters horizontally before reaching the edge of the cliff.

Step 2: Find the vertical distance the skier falls from the cliff.

Given:
- Height of the cliff (h): 30.0 m

The vertical distance fallen by the skier will be equal to the height of the cliff, so:
Vertical distance (y) = 30.0 m

Step 3: Find the total horizontal distance the skier travels.

To find the total horizontal distance (d_total), we can use the Pythagorean theorem.

Using the equation: d_total^2 = x^2 + y^2

Substituting the given values:
d_total^2 = (25.1 m)^2 + (30.0 m)^2
d_total^2 = 630.01 m^2 + 900 m^2
d_total^2 ≈ 1530.01 m^2

Taking the square root of both sides:
d_total ≈ √(1530.01) m
d_total ≈ 39.14 m

Therefore, the skier lands approximately 39.14 meters away from the bottom of the cliff.

To find how far from the bottom of the cliff the skier lands, we need to break down the problem into components and solve them step by step.

Step 1: Find the time taken by the skier to reach the edge of the cliff.
The skier starts from rest, so the initial velocity (u) is 0 m/s. The acceleration (a) is given as 4.00 m/s^2. We can use the equation of motion: s = ut + (1/2)at^2 to determine the time taken (t) to cover a distance of 50.0 m.
Substituting the values into the equation:
50.0m = (0)(t) + (1/2)(4.00m/s^2)(t^2)
50.0m = (1/2)(4.00m/s^2)(t^2)
100.0m = 4.00m/s^2 * t^2
t^2 = 100.0m / 4.00m/s^2
t^2 = 25.0s^2
t = √25.0s^2
t = 5.00s

Step 2: Calculate the horizontal distance traveled by the skier.
The horizontal distance traveled can be found using the equation: distance = speed × time. The speed in the horizontal direction remains constant throughout, and it is given by v = u + at. As the skier starts from rest, the initial horizontal velocity (u) is 0 m/s. Thus, the horizontal distance (d) is:
d = (u + at) × t
d = (0 + 4.00m/s^2 × 5.00s) × 5.00s
d = (20.0m/s) × 5.00s
d = 100.0m

Step 3: Calculate the vertical distance the skier drops.
The vertical distance the skier drops is equal to the height of the cliff, which is given as 30.0 m.

Step 4: Find the total distance from the bottom of the cliff where the skier lands.
The total distance from the bottom of the cliff where the skier lands is the hypotenuse of the right triangle formed by the horizontal distance traveled (d) and the vertical distance dropped (h), according to the Pythagorean theorem:
distance = √(d^2 + h^2)
distance = √(100.0m^2 + 30.0m^2)
distance = √(10000m^2 + 900m^2)
distance = √(10900m^2)
distance ≈ 104.4m (rounded to one decimal place)

Therefore, the skier lands approximately 104.4 meters from the bottom of the cliff.