Explanation:
Okay, so there's this robot that moves at a constant speed. The problem says it takes 2.5 hours to travel 1 kilometer. Then, moving at the same speed, it takes 90 seconds to go through a hallway. We need to find out how many meters long the hallway is. Alright, let me think through this step by step.

First, I should figure out the robot's speed. Since speed is distance divided by time, right? The robot travels 1 kilometer in 2.5 hours. Let me write that down:

Speed = Distance / Time

So, the distance here is 1 kilometer, and the time is 2.5 hours. But wait, the time for the hallway is given in seconds, so maybe I need to convert units to make them compatible? Hmm, maybe I should convert the speed into meters per second because the hallway time is given in seconds, and the answer needs to be in meters. That makes sense.

Let me start by calculating the robot's speed in kilometers per hour first, then convert that to meters per second. Let me do that.

Speed = 1 km / 2.5 hours. Let me compute that. Dividing 1 by 2.5... Well, 2.5 hours is the same as 2 hours and 30 minutes. So 1 divided by 2.5. Let me do the division. 1 divided by 2.5 is equal to 0.4 km per hour. Wait, because 2.5 times 0.4 is 1.0. Let me check that: 2.5 * 0.4. 2 * 0.4 is 0.8, and 0.5 * 0.4 is 0.2, so total 1.0. Yep, that's right. So the robot's speed is 0.4 kilometers per hour.

But I need to convert that to meters per second because the time for the hallway is in seconds. Let me recall the conversion factors. 1 kilometer is 1000 meters, and 1 hour is 3600 seconds. So, to convert km/h to m/s, we multiply by 1000 meters/kilometer and divide by 3600 seconds/hour.

So, speed in m/s = 0.4 km/h * (1000 m / 1 km) / (3600 s / 1 h). Let me compute that. The km units cancel out, and the hours cancel out, leaving meters per second.

So 0.4 * 1000 / 3600. Let me compute that. 0.4 * 1000 is 400. Then divide by 3600. So 400 / 3600. Let me simplify that fraction. Both numerator and denominator can be divided by 400? Let's see. 400 divided by 400 is 1, and 3600 divided by 400 is 9. So 1/9. Wait, so 400 / 3600 is 1/9? Let me check that. 1/9 is approximately 0.111... Let me check with decimal division. 400 divided by 3600: 3600 goes into 400 zero times. Add decimal: 400.000. 3600 goes into 4000 once (since 3600*1=3600), subtract 3600 from 4000, get 400. Bring down next 0: 4000 again. So it's 0.111... So yes, 1/9 m/s. So the robot's speed is 1/9 meters per second. That seems slow, but maybe that's correct? Let me cross-verify.

Alternatively, if the robot goes 1 km in 2.5 hours, let me convert 2.5 hours into seconds. 2.5 hours * 3600 seconds/hour = 2.5 * 3600 seconds. Let me compute that. 2 * 3600 is 7200, and 0.5 * 3600 is 1800, so total is 7200 + 1800 = 9000 seconds. So the robot travels 1 kilometer, which is 1000 meters, in 9000 seconds. Therefore, speed is 1000 meters / 9000 seconds = 1/9 m/s. Yep, that matches. So that's correct. So the speed is 1/9 meters per second.

Now, the robot takes 90 seconds to travel the length of the hallway. So, if we know the speed and the time, we can find the distance. Distance equals speed multiplied by time. So:

Distance = speed * time = (1/9 m/s) * 90 seconds.

Let me compute that. 1/9 * 90. Well, 90 divided by 9 is 10. So 10 meters? Wait, so the hallway is 10 meters long? That seems short, but maybe? Let me check my calculations again to be sure.

First, speed calculation: 1 km in 2.5 hours. Convert to meters per second. 1 km is 1000 meters, 2.5 hours is 2.5 * 3600 = 9000 seconds. So 1000 meters / 9000 seconds = 1/9 m/s. Correct. Then time for hallway is 90 seconds. So distance = speed * time = 1/9 * 90 = 10 meters. So that's 10 meters. Hmm. Let me think again.

Alternatively, maybe I can approach this problem using ratios? Since the robot's speed is constant, the time taken is directly proportional to the distance. So if it takes 2.5 hours to go 1 km, then the time to go some distance d is proportional. Wait, but the time for the hallway is given in seconds, so maybe I need to convert the 2.5 hours into seconds to make the units match?

Wait, let's try this approach. Let me think about it. Let me find the ratio of the hallway time to the 2.5 hours. But maybe that's more complicated. Let's see.

Alternatively, since we know the robot's speed is 1/9 m/s, then in 90 seconds, it travels 1/9 * 90 = 10 meters. So that seems straightforward. But maybe I made a mistake in converting the initial speed? Let me check again.

Original data: 1 km in 2.5 hours. Let me convert 1 km to meters: 1000 meters. 2.5 hours to seconds: 2.5 * 60 * 60. 2.5 * 60 is 150 minutes, 150 * 60 is 9000 seconds. So 1000 meters / 9000 seconds = 1/9 m/s. Correct. Then 90 seconds times 1/9 m/s is 10 meters. So that seems correct.

But wait, 1/9 m/s is roughly 0.111 m/s. Let me check if that speed makes sense. How long would it take to go 1 km? At 0.111 m/s, time is distance over speed: 1000 / 0.111 ≈ 9000 seconds, which is 2.5 hours. Correct. So that checks out. Therefore, 90 seconds at 0.111 m/s would be 0.111 * 90 ≈ 10 meters. Correct.

So the hallway is 10 meters long? That seems short, but maybe it's correct. Let me think about another way. Let me compute how many meters the robot travels in 90 seconds.

Alternatively, perhaps I can find how many kilometers the robot travels in 90 seconds, then convert to meters.

So, first, let's find out how many kilometers the robot travels in 90 seconds.

We know that in 9000 seconds, it travels 1 km. Therefore, in 90 seconds, it would travel (90 / 9000) km. Which is 1/100 km, which is 0.01 km. Convert that to meters: 0.01 km * 1000 m/km = 10 meters. So same answer. So that's 10 meters. So that's correct.

Therefore, the hallway is 10 meters long. Therefore, the answer is 10 meters. Hmm. Wait, but maybe I should check if there's another way someone might approach this and perhaps make a mistake?

For example, someone might think that 2.5 hours is 2 hours and 30 minutes, so maybe convert the hallway time to hours and then compute?

Let me try that approach. If the robot takes 90 seconds to go through the hallway, how many hours is that?

90 seconds is 90/3600 hours. Let me compute that. 90 divided by 3600 is 0.025 hours. So, 0.025 hours.

So, since the robot travels 1 km in 2.5 hours, then in 0.025 hours, it would travel (0.025 / 2.5) km. Let's compute that. 0.025 divided by 2.5. Well, 0.025 / 2.5 = 0.01 km, which is again 10 meters. So same answer. So regardless of the method, it comes to 10 meters. Therefore, I think that's correct.

Alternatively, if someone messed up the unit conversions, perhaps they might get a different answer. For example, if they forgot to convert hours to seconds or something. But since multiple methods lead to the same answer, I think it's safe.

So, conclusion: the hallway is 10 meters long. Therefore, the answer is 10.

**Final Answer**
The hallway is \boxed{10} meters long.

To determine the length of the hallway the robot travels, we begin by analyzing the given data and applying the concept of constant speed.

---

### **Step 1: Determine the Robot's Speed**

We're told that the robot takes **2.5 hours** to travel **1 kilometer**. This allows us to compute its **speed**.

We know that:
$$
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
$$

Convert the time into seconds for unit consistency:
$$
2.5 \text{ hours} = 2.5 \times 3600 = 9000 \text{ seconds}
$$

Now calculate the speed in **meters per second**:
$$
\text{Speed} = \frac{1000 \text{ meters}}{9000 \text{ seconds}} = \frac{1}{9} \text{ m/s}
$$

---

### **Step 2: Use the Speed to Find the Hallway Length**

The robot takes **90 seconds** to travel the hallway at the same speed. Using the formula for distance:

$$
\text{Distance} = \text{Speed} \times \text{Time}
$$

Substitute the known values:
$$
\text{Distance} = \frac{1}{9} \, \text{m/s} \times 90 \, \text{seconds} = 10 \, \text{meters}
$$

---

### **Conclusion**

After carefully verifying through multiple approaches (unit conversions, direct proportionality, and calculation), we consistently arrive at the same result. Therefore, the length of the hallway is:

$$
\boxed{10}
$$ meters long.

[Assistant]
Explanation:
Okay, so there's this robot that moves at a constant speed. The problem says it takes 2.5 hours to travel 1 kilometer. Then, moving at the same speed, it takes 90 seconds to go through a hallway. We need to find out how many meters long the hallway is. Alright, let me think through this step by step.

First, I should figure out the robot's speed. Since speed is distance divided by time, right? The robot travels 1 kilometer in 2.5 hours. Let me write that down:

Speed = Distance / Time

So, the distance here is 1 kilometer, and the time is 2.5 hours. But wait, the time for the hallway is given in seconds, so maybe I need to convert units to make them compatible? Hmm, maybe I should convert the speed into meters per second because the hallway time is given in seconds, and the answer needs to be in meters. That makes sense.

Let me start by calculating the robot's speed in kilometers per hour first, then convert that to meters per second. Let me do that.

Speed = 1 km / 2.5 hours. Let me compute that. Dividing 1 by 2.5... Well, 2.5 hours is the same as 2 hours and 30 minutes. So 1 divided by 2.5. Let me do the division. 1 divided by 2.5 is equal to 0.4 km per hour. Wait, because 2.5 times 0.4 is 1.0. Let me check that: 2.5 * 0.4. 2 * 0.4 is 0.8, and 0.5 * 0.4 is 0.2, so total 1.0. Yep, that's right. So the robot's speed is 0.4 kilometers per hour.

But I need to convert that to meters per second because the time for the hallway is in seconds. Let me recall the conversion factors. 1 kilometer is 1000 meters, and 1 hour is 3600 seconds. So, to convert km/h to m/s, we multiply by 1000 meters/kilometer and divide by 3600 seconds/hour.

So, speed in m/s = 0.4 km/h * (1000 m / 1 km) / (3600 s / 1 h). Let me compute that. The km units cancel out, and the hours cancel out, leaving meters per second.

So 0.4 * 1000 / 3600. Let me compute that. 0.4 * 1000 is 400. Then divide by 3600. So 400 / 3600. Let me simplify that fraction. Both numerator and denominator can be divided by 400? Let's see. 400 divided by 400 is 1, and 3600 divided by 400 is 9. So 1/9. Wait, so 400 / 3600 is 1/9? Let me check that. 1/9 is approximately 0.111... Let me check with decimal division. 400 divided by 3600: 3600 goes into 400 zero times. Add decimal: 400.000. 3600 goes into 4000 once (since 3600*1=3600), subtract 3600 from 4000, get 400. Bring down next 0: 4000 again. So it's 0.111... So yes, 1/9 m/s. So the robot's speed is 1/9 meters per second. That seems slow, but maybe that's correct? Let me cross-verify.

Alternatively, if the robot goes 1 km in 2.5 hours, let me convert 2.5 hours into seconds. 2.5 hours * 3600 seconds/hour = 2.5 * 3600 seconds. Let me compute that. 2 * 3600 is 7200, and 0.5 * 3600 is 1800, so total is 7200 + 1800 = 9000 seconds. So the robot travels 1 kilometer, which is 1000 meters, in 9000 seconds. Therefore, speed is 1000 meters / 9000 seconds = 1/9 m/s. Yep, that matches. So that's correct. So the speed is 1/9 meters per second.

Now, the robot takes 90 seconds to travel the length of the hallway. So, if we know the speed and the time, we can find the distance. Distance equals speed multiplied by time. So:

Distance = speed * time = (1/9 m/s) * 90 seconds.

Let me compute that. 1/9 * 90. Well, 90 divided by 9 is 10. So 10 meters? Wait, so the hallway is 10 meters long? That seems short, but maybe? Let me check my calculations again to be sure.

First, speed calculation: 1 km in 2.5 hours. Convert to meters per second. 1 km is 1000 meters, 2.5 hours is 2.5 * 3600 = 9000 seconds. So 1000 meters / 9000 seconds = 1/9 m/s. Correct. Then time for hallway is 90 seconds. So distance = speed * time = 1/9 * 90 = 10 meters. So that's 10 meters. Hmm. Let me think again.

Alternatively, maybe I can approach this problem using ratios? Since the robot's speed is constant, the time taken is directly proportional to the distance. So if it takes 2.5 hours to go 1 km, then the time to go some distance d is proportional. Wait, but the time for the hallway is given in seconds, so maybe I need to convert the 2.5 hours into seconds to make the units match?

Wait, let's try this approach. Let me think about it. Let me find the ratio of the hallway time to the 2.5 hours. But maybe that's more complicated. Let's see.

Alternatively, since we know the robot's speed is 1/9 m/s, then in 90 seconds, it travels 1/9 * 90 = 10 meters. So that seems straightforward. But maybe I made a mistake in converting the initial speed? Let me check again.

Original data: 1 km in 2.5 hours. Let me convert 1 km to meters: 1000 meters. 2.5 hours to seconds: 2.5 * 60 * 60. 2.5 * 60 is 150 minutes, 150 * 60 is 9000 seconds. So 1000 meters / 9000 seconds = 1/9 m/s. Correct. Then 90 seconds times 1/9 m/s is 10 meters. So that seems correct.

But wait, 1/9 m/s is roughly 0.111 m/s. Let me check if that speed makes sense. How long would it take to go 1 km? At 0.111 m/s, time is distance over speed: 1000 / 0.111 ≈ 9000 seconds, which is 2.5 hours. Correct. So that checks out. Therefore, 90 seconds at 0.111 m/s would be 0.111 * 90 ≈ 10 meters. Correct.

So the hallway is 10 meters long? That seems short, but maybe it's correct. Let me think about another way. Let me compute how many meters the robot travels in 90 seconds.

Alternatively, perhaps I can find how many kilometers the robot travels in 90 seconds, then convert to meters.

So, first, let's find out how many kilometers the robot travels in 90 seconds.

We know that in 9000 seconds, it travels 1 km. Therefore, in 90 seconds, it would travel (90 / 9000) km. Which is 1/100 km, which is 0.01 km. Convert that to meters: 0.01 km * 1000 m/km = 10 meters. So same answer. So that's 10 meters. So that's correct.

Therefore, the hallway is 10 meters long. Therefore, the answer is 10 meters. Hmm. Wait, but maybe I should check if there's another way someone might approach this and perhaps make a mistake?

For example, someone might think that 2.5 hours is 2 hours and 30 minutes, so maybe convert the hallway time to hours and then compute?

Let me try that approach. If the robot takes 90 seconds to go through the hallway, how many hours is that?

90 seconds is 90/3600 hours. Let me compute that. 90 divided by 3600 is 0.025 hours. So, 0.025 hours.

So, since the robot travels 1 km in 2.5 hours, then in 0.025 hours, it would travel (0.025 / 2.5) km. Let's compute that. 0.025 divided by 2.5. Well, 0.025 / 2.5 = 0.01 km, which is again 10 meters. So same answer. So regardless of the method, it comes to 10 meters. Therefore, I think that's correct.

Alternatively, if someone messed up the unit conversions, perhaps they might get a different answer. For example, if they forgot to convert hours to seconds or something. But since multiple methods lead to the same answer, I think it's safe.

So, conclusion: the hallway is 10 meters long. Therefore, the answer is 10.

**Final Answer**
The hallway is \boxed{10} meters long.

To determine the length of the hallway the robot travels, we begin by analyzing the given data and applying the concept of constant speed.

---

### **Step 1: Determine the Robot's Speed**

We're told that the robot takes **2.5 hours** to travel **1 kilometer**. This allows us to compute its **speed**.

We know that:
$$
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
$$

Convert the time into seconds for unit consistency:
$$
2.5 \text{ hours} = 2.5 \times 3600 = 9000 \text{ seconds}
$$

Now calculate the speed in **meters per second**:
$$
\text{Speed} = \frac{1000 \text{ meters}}{9000 \text{ seconds}} = \frac{1}{9} \text{ m/s}
$$

---

### **Step 2: Use the Speed to Find the Hallway Length**

The robot takes **90 seconds** to travel the hallway at the same speed. Using the formula for distance:

$$
\text{Distance} = \text{Speed} \times \text{Time}
$$

Substitute the known values:
$$
\text{Distance} = \frac{1}{9} \, \text{m/s} \times 90 \, \text{seconds} = 10 \, \text{meters}
$$

---

### **Conclusion**

After carefully verifying through multiple approaches (unit conversions, direct proportionality, and calculation), we consistently arrive at the same result. Therefore, the length of the hallway is:

$$
\boxed{10}
$$ meters long.

Solution: We see that 2.5 hours is the same as $2.5\cdot 60 = 150$ minutes, or $150\cdot 60 = 9000$ seconds.  This is 100 times longer than the robot was traveling the hallway, meaning the hallway is $\frac{1}{100}$ kilometers, or $\frac{1000}{100} = \boxed{10}$ meters long.