Okay, let's try to figure out this problem. So, we have an alien civilization in the Large Magellanic Cloud, and they built a spacecraft that can travel at 0.99999987 times the speed of light. The question is about how long it will take for a 22-year-old astronaut to reach Earth from their perspective. The options are A to D, with A suggesting the astronaut dies before reaching, and the others giving years in the 70s and 80s.

First, I remember that when dealing with high speeds close to the speed of light, time dilation and length contraction are significant factors. So, the astronaut's experienced time will be different from the Earth's time. But wait, the question is asking from the astronaut's point of view, so we need to consider how much time passes for them during the journey.

Let me start by calculating the distance to Earth from the Large Magellanic Cloud. Wait, the Large Magellanic Cloud is a distant galaxy, but the problem doesn't specify the exact distance. Hmm, maybe I need to make an assumption here. I recall that the average distance from Earth to the Large Magellanic Cloud is about 160,000 light-years. Let me confirm that. Yes, that's approximately 160,000 light-years. So, the distance is 1.6e5 light-years.

The spacecraft's speed is 0.99999987c. Let me write that as v = 0.99999987c. To find the time from Earth's perspective, we can use time = distance/speed. But wait, the astronaut's time will be different because of time dilation. However, since the question is from the astronaut's perspective, we need to calculate the proper time experienced by the astronaut.

Proper time (τ) is calculated using the time dilation formula: τ = t * sqrt(1 - v²/c²). But here, t is the time measured in Earth's frame. Alternatively, another way is to use the distance as seen from the spacecraft's frame. Wait, no. Let me think again. The proper time is the time experienced by the astronaut, who is in the spacecraft. So, the proper time is given by τ = t * sqrt(1 - v²/c²), where t is the time in Earth's frame.

But first, let's compute the time from Earth's perspective. The distance is 1.6e5 light-years, and the speed is 0.99999987c. So, time t = distance / speed = (1.6e5 light-years) / (0.99999987c). Since speed is in terms of c, and distance is in light-years, the units will cancel out. Let's compute that.

First, 1 light-year at speed c takes 1 year. So, time t = 1.6e5 / 0.99999987. Let me compute that. 1.6e5 divided by 0.99999987. Since 0.99999987 is very close to 1, this will be slightly more than 1.6e5. Let me approximate it. 0.99999987 is 1 - 1.3e-8. So, 1/0.99999987 ≈ 1 + 1.3e-8. Therefore, t ≈ 1.6e5 * (1 + 1.3e-8) ≈ 1.6e5 + 2.08e-3. So, approximately 160,000.00208 years. But the difference here is negligible, so maybe we can approximate t ≈ 1.6e5 years. But actually, since the speed is very close to c, the time from Earth's perspective would be approximately 1.6e5 / 0.99999987, which is slightly more than 160,000 years. Let me compute it more accurately.

Let me compute 1.6e5 divided by 0.99999987. Let's write 0.99999987 as 1 - 1.3e-8. So, dividing by (1 - x) where x is 1.3e-8. Using the approximation 1/(1 - x) ≈ 1 + x for small x, so t ≈ 1.6e5 * (1 + 1.3e-8) ≈ 1.6e5 + 1.6e5 * 1.3e-8. Calculating 1.6e5 * 1.3e-8: 1.6 * 1.3 = 2.08, and 1e5 * 1e-8 = 1e-3. So, 2.08e-3. Therefore, t ≈ 160,000 + 0.00208 ≈ 160,000.00208 years. So, very slightly more than 160,000 years. But since the options are in the 70s and 80s, maybe the Earth's time is not the way to go. Wait, the question is about the astronaut's time. So, the proper time experienced by the astronaut would be shorter due to time dilation. The formula is τ = t * sqrt(1 - v²/c²). Let's compute sqrt(1 - v²/c²).

v = 0.99999987c, so v²/c² = (0.99999987)^2. Let me compute that. Let's compute (1 - 1.3e-8)^2. Using the binomial expansion: (1 - 2*1.3e-8 + (1.3e-8)^2). The last term is negligible, so approximately 1 - 2.6e-8. Therefore, 1 - v²/c² ≈ 2.6e-8. So, sqrt(1 - v²/c²) ≈ sqrt(2.6e-8). Let's compute that. sqrt(2.6e-8) = sqrt(2.6) * 1e-4 ≈ 1.612 * 1e-4 = 1.612e-4. So, τ ≈ t * 1.612e-4. But t is approximately 160,000 years. So, τ ≈ 160,000 * 1.612e-4 ≈ 160,000 * 0.0001612. Let's compute that. 160,000 * 0.0001612 = 160,000 * 1.612e-4. 160,000 * 1e-4 = 16, so 16 * 1.612 = 25.792 years. Wait, that's about 25.8 years. But the options are 72, 77, 81, or A. So, this suggests that the astronaut would age approximately 25.8 years, so the total time experienced would be around 26 years. But the options don't have that. So, maybe I made a mistake here.

Wait, perhaps I need to calculate the proper time using the distance in the spacecraft's frame. Because in the spacecraft's frame, the distance is contracted. So, the proper length (distance in the spacecraft's frame) is L = L0 * sqrt(1 - v²/c²), where L0 is the distance in Earth's frame. So, L = 1.6e5 light-years * sqrt(1 - v²/c²). Then, the time experienced by the astronaut is L / v. Let's try this approach.

First, compute sqrt(1 - v²/c²) again. As before, v = 0.99999987c, so v²/c² = (0.99999987)^2 ≈ 1 - 2*1.3e-8 + (1.3e-8)^2 ≈ 1 - 2.6e-8. So, sqrt(1 - v²/c²) ≈ sqrt(2.6e-8) ≈ 1.612e-4. So, the contracted length L = 1.6e5 ly * 1.612e-4 ≈ 1.6e5 * 1.612e-4. Let's compute that: 1.6 * 1.612 ≈ 2.5792, and 1e5 * 1e-4 = 1e1. So, 2.5792 * 10 = 25.792 light-years. So, the spacecraft sees the distance as about 25.792 light-years. Then, the time experienced by the astronaut is L / v = 25.792 ly / (0.99999987c). Since speed is c, time is 25.792 / 0.99999987 years. Let's compute that. 25.792 / 0.99999987 ≈ 25.792 * (1 + 1.3e-8) ≈ 25.792 + 25.792 * 1.3e-8. The second term is negligible, so approximately 25.792 years. So, the astronaut's time is about 25.8 years. But again, the options don't have that. So, perhaps I made a mistake in the calculation.

Wait, maybe the problem is that the lifetime of the aliens is 150 solar years. But the astronaut is 22 years old. So, they are traveling to Earth, which is 160,000 light-years away. But the spacecraft's speed is 0.99999987c. Let me check the calculation again.

Alternatively, perhaps I should compute the time from Earth's frame and then apply time dilation. Let's see. The time from Earth's frame is t = distance / speed = 1.6e5 ly / 0.99999987c. Since speed is in terms of c, and distance is in ly, the time is (1.6e5) / 0.99999987 years. Let's compute that. 1.6e5 / 0.99999987 ≈ 160,000 * (1 + 1.3e-8) ≈ 160,000 + 160,000 * 1.3e-8 ≈ 160,000 + 0.00208 ≈ 160,000.00208 years. So, approximately 160,000 years. Then, the proper time experienced by the astronaut is τ = t * sqrt(1 - v²/c²) ≈ 160,000 * 1.612e-4 ≈ 25.792 years. Again, same result. But the options are way higher. So, perhaps there's a misunderstanding in the problem.

Wait, maybe the problem is not about time dilation, but about the fact that the spacecraft's speed is so high that the Earth is moving towards the spacecraft at that speed. But that's more about relative velocity, but since the spacecraft is moving at 0.99999987c, the Earth would be moving towards it at almost the same speed. However, the distance is still the same, so the time experienced by the astronaut is still the contracted time.

Alternatively, perhaps I need to consider the Earth's time as the coordinate time, and the astronaut's time as the proper time. But the calculation shows that the proper time is about 25.8 years. But the options are 72, 77, 81, or A. So, maybe my approach is wrong.

Wait, another thought: maybe the problem is considering the time from the perspective of the Earth, but the astronaut's life is 150 years. Wait, but the astronaut is only 22 years old. So, if the journey takes 160,000 years from Earth's perspective, the astronaut would age 22 + 160,000 years, which is obviously impossible. But that's not the case. The problem states that the aliens have an average lifetime of 150 years. Wait, but the astronaut is 22 years old, so if the journey takes, say, 160,000 years from Earth's perspective, the astronaut would live for 150 years, but the journey would take much longer. But that's not relevant here. The question is about the astronaut's experienced time. So, the proper time is around 25.8 years. But the options are much larger. So, perhaps I'm missing something.

Wait, perhaps the problem is not considering the distance to Earth correctly. Let me check the distance again. The Large Magellanic Cloud is a galaxy, and the distance to Earth is roughly 160,000 light-years. But maybe in the problem, the distance is given as something else. Wait, the problem doesn't specify the distance. Hmm. Maybe the distance is not 160,000 light-years but something else. Maybe the problem assumes that the distance is 160,000 ly, but perhaps the speed is 0.99999987c, so maybe the time from Earth's frame is 160,000 / 0.99999987 ≈ 160,000.002 years. But the astronaut's proper time is 25.8 years. But the options are 72, 77, 81. So, perhaps there's a miscalculation here.

Wait, maybe I made a mistake in the calculation of sqrt(1 - v²/c²). Let me recalculate that. Let's compute v = 0.99999987c. So, v²/c² = (0.99999987)^2. Let's compute that more accurately. Let me write 0.99999987 as 1 - 0.00000013. Wait, 1 - 0.00000013 = 0.99999987? Let me check: 0.00000013 is 1.3e-7, but 0.99999987 is 1 - 1.3e-8, right? Wait, 1.3e-8 is 0.000000013, so 1 - 0.000000013 = 0.999999987. Wait, but the given speed is 0.99999987c, which is 1 - 0.00000013. Wait, 0.99999987 is 1 - 0.00000013. So, v = 1 - 1.3e-7. Wait, 0.00000013 is 1.3e-7. So, v = 1 - 1.3e-7. Then, v² = (1 - 1.3e-7)^2 = 1 - 2*1.3e-7 + (1.3e-7)^2 ≈ 1 - 2.6e-7. So, 1 - v²/c² ≈ 2.6e-7. Then, sqrt(1 - v²/c²) ≈ sqrt(2.6e-7) ≈ 5.1e-4. So, sqrt(1 - v²/c²) ≈ 0.00051. Then, the proper time τ = t * 0.00051. But t is 160,000 / 0.99999987 ≈ 160,000.002 years. So, τ ≈ 160,000 * 0.00051 ≈ 81.6 years. Oh! Wait, that's approximately 81.6 years. So, the astronaut's time would be about 81.6 years. Then, adding their age, they were 22, so 22 + 81.6 ≈ 103.6 years. But the question is asking how long it will take for their 22-year-old astronaut to reach Earth. So, the time experienced by the astronaut is 81.6 years. So, the answer would be approximately 81 years, which is option C. But wait, let me double-check the calculation.

Wait, earlier I thought v = 0.99999987c is 1 - 1.3e-8, but actually, 0.99999987 is 1 - 0.00000013, which is 1.3e-7. So, the correct expansion is v = 1 - 1.3e-7. Then, v² = (1 - 1.3e-7)^2 = 1 - 2*1.3e-7 + (1.3e-7)^2 ≈ 1 - 2.6e-7. Then, 1 - v²/c² ≈ 2.6e-7. So, sqrt(2.6e-7) ≈ 5.1e-4. Therefore, τ = t * 5.1e-4. And t is 160,000 / 0.99999987 ≈ 160,000.002 years. So, τ ≈ 160,000 * 5.1e-4 ≈ 81.6 years. So, that's approximately 81 years. So, the answer would be C. 81 years.

But wait, the options are A: die before, B:72, C:81, D:77. So, 81 is option C. But let me check if the calculation is correct. Let me compute 1 - v²/c² more accurately. Let's compute (0.99999987)^2. Let's compute 0.99999987 * 0.99999987. Let's do this step by step.

First, 1 * 1 = 1.

Now, subtract 0.00000013 from 1 twice, but that's complicated. Alternatively, use (1 - a)^2 = 1 - 2a + a², where a = 0.00000013. So, (0.99999987)^2 = (1 - 1.3e-7)^2 = 1 - 2*(1.3e-7) + (1.3e-7)^2. The third term is negligible. So, 1 - 2.6e-7. Therefore, 1 - v²/c² = 2.6e-7. So, sqrt(2.6e-7) = sqrt(2.6)*1e-3.5 ≈ 1.612*1e-3.5. Wait, 1e-7 is (1e-3.5)^2. So, sqrt(2.6e-7) = sqrt(2.6)*1e-3.5 ≈ 1.612*0.0000316 ≈ 0.000051. So, approximately 5.1e-5. Wait, wait, 1e-7 is (1e-3.5)^2. Wait, 1e-3.5 is 3.16e-4, so (3.16e-4)^2 = 1e-8. Wait, I'm getting confused. Let me compute sqrt(2.6e-7):

sqrt(2.6e-7) = sqrt(2.6) * sqrt(1e-7) = 1.612 * (1e-7)^0.5 = 1.612 * 3.16e-4 ≈ 1.612 * 0.000316 ≈ 0.00051. So, approximately 0.00051. So, τ = t * 0.00051.

Now, t = 1.6e5 ly / 0.99999987c ≈ 160,000 / 0.99999987 ≈ 160,000.002 years. So, τ ≈ 160,000 * 0.00051 ≈ 81.6 years. So, that's about 81.6 years. So, the astronaut would age approximately 81.6 years, so the total time experienced by the astronaut is about 81.6 years. Therefore, the answer should be C. 81 years.

But wait, the options are 72, 77, 81, or A. So, 81 is option C. But let me check if there's any other factor I might have missed. For example, the time dilation factor is sqrt(1 - v²/c²), which is approximately 0.00051, leading to τ ≈ 81.6 years. So, that's the correct calculation. Therefore, the answer is C. 81 years.
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