Final answer:
The sister traveling at 0.8c for 20 years according to her brother on Earth will be 2 years older than her brother when she returns. This follows from the use of the Lorentz factor to calculate the time dilation effect for her trip.
Step-by-step explanation:
The question involves the concept known as the Twin Paradox, which is a consequence of time dilation described by the theory of relativity. When the 30-year-old sister travels at a speed of 0.8c for 20 years according to her 20-year-old brother on Earth, we use the Lorentz factor (also denoted as 'gamma' or 'y') to calculate the time experienced by the traveling sister. The Lorentz factor is calculated using the equation y = 1 / sqrt(1 - (v^2 / c^2)), where 'v' is the velocity and 'c' is the speed of light. For 0.8c, we have y ≈ 1.6667.
To find out how much time has passed for the traveling sister, we divide the 20 years by y: 20 years / 1.6667 ≈ 12 years. Therefore, when the traveling sister returns, she will be 30 years + 12 years = 42 years old, while her brother will be 20 years + 20 years = 40 years old. Hence, the sister will be 2 years older than her brother, which is not one of the provided options in the multiple-choice question. Typically, this kind of discrepancy may be due to a problem in the phrasing of the question or the options provided.