Final answer:
The speed of the traveling wave on the climbing rope at its midpoint, calculated using the provided density and the known equation for wave speed on a rope, is approximately 18.43 m/s.
None of the given options correspond to this calculated value.
Step-by-step explanation:
The speed of a traveling wave on a rope depends on the tension in the rope and the mass per unit length (linear density) of the rope.
The formula for the speed (v) of a wave on a stretched string is v = sqrt(T/μ), where (T) is the tension and (μ) is the linear density. Since we know the density of the climbing rope is 0.05 kg/m, we can calculate the mass of the rope hanging from the midpoint as half the length of the rope times the density.
Therefore, the mass at the midpoint is (0.05 kg/m) * (70 m / 2) = 1.75 kg. The tension at the midpoint due to the weight of the bottom half of the rope will be the mass times the acceleration due to gravity (T = m * g). Thus, T = 1.75 kg * 9.81 m/s^2 = 17.1675 N. Using the wave speed formula, we calculate speed as v = sqrt(17.1675 N / 0.05 kg/m), which when computed gives v ~= 18.43 m/s.
However, this result is not one of the options given, indicating that perhaps the options may contain a typo or the question might have an error. All provided options (7 m/s, 10 m/s, 14 m/s, 20 m/s) do not match the calculated value. If the question itself is correct, there could be a misunderstanding in the information provided or a mistake in the calculation.