To solve this problem, we can use conservation of energy:
Initial mechanical energy (at the top of the hill) = final mechanical energy (at the bottom of the hill)
Initial mechanical energy = mgh, where m is the total mass of the child and sled, g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height of the hill. Since the hill makes an angle of 16.9° with respect to the horizontal, we can use trigonometry to find the height:
h = (length of slope) * sin(16.9°)
We don't know the length of the slope, but we can solve for it by using the final speed of the child and sled:
Final mechanical energy = (1/2)mv^2, where v is the final speed of the child and sled.
Setting these equal, we get:
mgh = (1/2)mv^2
Substituting for h and solving for the length of the slope:
(length of slope) = v^2 / (2g*sin(16.9°))
Plugging in the given values:
(length of slope) = (6.8 m/s)^2 / (29.8 m/s^2sin(16.9°)) ≈ 12.5 meters
Therefore, the child and sled slide about 12.5 meters along the slope.