A girl is sledding down a slope that is inclined at 30º with respect to the horizontal. The wind is aiding the motion by providing a steady force of 131 N that is parallel to the motion of the sled. The combined mass of the girl and the sled is 59.9 kg, and the coefficient of kinetic friction between the snow and the runners of the sled is 0.217. How much time is required for the sled to travel down a 243-m slope, starting from rest?

To solve the problem, we need to break it down into steps and use a combination of Newton's laws of motion and kinematic equations.

Step 1: Determine the forces acting on the sled.
In this case, there are three forces acting on the sled:
1. The force of gravity pulling the sled downwards, which can be calculated using the formula Fg = m * g, where m is the mass of the sled and g is the acceleration due to gravity (9.8 m/s²).
2. The component of the force of gravity acting parallel to the slope, which is calculated as Fg_parallel = Fg * sin(θ), where θ is the angle of inclination (30º in this case).
3. The force provided by the wind, which is given as 131 N and is parallel to the motion of the sled.

Step 2: Calculate the net force acting on the sled.
The net force is the vector sum of all the forces acting on the sled. Since all the forces are acting in the same direction (downhill), we can add them up to get the net force. The net force is calculated as Fnet = Fg_parallel + F_wind.

Step 3: Calculate the acceleration of the sled.
Using Newton's second law, which states that Fnet = m * a, we can plug in the values of Fnet (obtained from step 2) and mass (given as 59.9 kg) to calculate the acceleration of the sled.

Step 4: Calculate the time required to travel down the slope.
To find the time required, we will use the third equation of motion: s = ut + (1/2)at², where s is the distance traveled, u is the initial velocity (which is 0 since the sled starts from rest), a is the acceleration (obtained from step 3), and t is the time.

Now, let's calculate the time required for the sled to travel down the slope:

Step 1: Determine the forces acting on the sled.
Fg = m * g = 59.9 kg * 9.8 m/s² = 586.02 N
Fg_parallel = Fg * sin(30º) = 586.02 N * sin(30º) = 293.01 N

Step 2: Calculate the net force acting on the sled.
Fnet = Fg_parallel + F_wind = 293.01 N + 131 N = 424.01 N

Step 3: Calculate the acceleration of the sled.
Fnet = m * a
424.01 N = 59.9 kg * a
a = 7.08 m/s²

Step 4: Calculate the time required to travel down the slope.
s = ut + (1/2)at²
243 m = 0 + (1/2) * 7.08 m/s² * t²
Using algebra, we can rearrange the equation to solve for t:
t² = (2 * 243 m) / 7.08 m/s²
t² = 68.64 s²
t ≈ 8.29 s

Therefore, it will take approximately 8.29 seconds for the sled to travel down the 243-meter slope starting from rest.