First, let's parse the given information. The total distance is 224 km, which is 224,000 meters. She divides this into four parts, so each part is a quarter of the total distance? Wait, no, not necessarily, because the speeds are in AP, so the distances might not be equal. Hmm, right, because speed and distance are related by time, and since time is also in AP, this complicates things.
She travels the first part at 960 meters per minute for 30 minutes. Calculate the distance for the first part. Speed is 960 m/min, time is 30 minutes, so distance is speed multiplied by time. That would be 960 * 30 = 28,800 meters. So the first part is 28,800 meters.
Now, the total distance is 224,000 meters, so the remaining three parts add up to 224,000 - 28,800 = 195,200 meters. Let me note that down.
Next, her speeds are in an arithmetic progression. Let's denote her speeds as v1, v2, v3, v4. Since it's an AP, the difference between consecutive terms is constant. Let's say the common difference is d. So, v1 = v, v2 = v + d, v3 = v + 2d, v4 = v + 3d.
We know that v1 is 960 m/min. So, v = 960 m/min.
Similarly, the times taken for each part are in another arithmetic progression. Let's denote the times as t1, t2, t3, t4. Since it's an AP, the difference between consecutive terms is constant. Let's say the common difference is k. So, t1 = t, t2 = t + k, t3 = t + 2k, t4 = t + 3k.
We know that t1 is 30 minutes. So, t = 30 minutes.
The total time is 3 hours, which is 180 minutes. So, t1 + t2 + t3 + t4 = 180 minutes.
Substituting the AP terms, we get:
t + (t + k) + (t + 2k) + (t + 3k) = 180
Simplifying:
4t + 6k = 180
We know t = 30, so:
4*30 + 6k = 180
120 + 6k = 180
6k = 60
k = 10 minutes.
So, the times are 30, 40, 50, 60 minutes.
If k = 10, then t1=30, t2=40, t3=50, t4=60. Adding them up: 30+40=70, 50+60=110, total 180. Yes, that works.
Now, we have the times for each part. The distance for each part is speed multiplied by time. So, distance1 = v1 t1 = 960 30 = 28,800 m, which we already have.
Distance2 = v2 t2 = (v + d) 40
Distance3 = v3 t3 = (v + 2d) 50
Distance4 = v4 t4 = (v + 3d) 60
We know the total distance is 224,000 m, so:
28,800 + 40(v + d) + 50(v + 2d) + 60(v + 3d) = 224,000
Compute each term:
40(v + d) = 40v + 40d
50(v + 2d) = 50v + 100d
60(v + 3d) = 60v + 180d
Adding all together:
28,800 + (40v + 40d) + (50v + 100d) + (60v + 180d) = 224,000
Combine like terms:
28,800 + (40v + 50v + 60v) + (40d + 100d + 180d) = 224,000
28,800 + 150v + 320d = 224,000
Subtract 28,800 from both sides:
150v + 320d = 224,000 - 28,800 = 195,200
We know v = 960, so plug that in:
150*960 + 320d = 195,200
Calculate 150*960:
150 * 960 = 144,000
So:
144,000 + 320d = 195,200
Subtract 144,000:
320d = 51,200
Divide both sides by 320:
d = 51,200 / 320
Calculate that:
51,200 ÷ 320 = 160
So, d = 160 m/min.
Wait, hold on, that can't be right. Because d is in m/min, but speed is in m/min, so adding 160 m/min to 960 m/min would make sense.
Yes, 150*960 is indeed 144,000. 195,200 - 144,000 is 51,200. 51,200 divided by 320 is 1
