Zhang drove from location A at 8:00 and traveled back and forth between A and B. 8: Accelerate by 30% when reaching point C between A and B for the first time, and continue driving until reaching point C for the second time at 9:00. How many times is the distance between AC and BC?
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Let's understand this problem step by step. Zhang starts driving from point A at 8:00 AM, traveling back and forth between points A and B. Point C is located somewhere between A and B. When Zhang reaches point C for the first time at 8:30, his speed increases by 30 percent. He then continues driving and reaches point C for the second time at 9:00 AM. We need to find the ratio of distances AC to BC.
Now let's set up our variables to solve this problem systematically. Let AC equal x, which is the distance from A to C. Let BC equal y, which is the distance from C to B. Zhang's initial speed is v, and after reaching C at 8:30, his speed increases by 30 percent to 1.3v. We know that it takes exactly 30 minutes to travel from A to C initially, and another 30 minutes to return to C after the speed increase.
Let's analyze Zhang's complete journey. In Phase 1, from 8:00 to 8:30, Zhang travels from A to C, covering distance x at speed v in 30 minutes. In Phase 2, from 8:30 to 9:00, Zhang travels from C to B and then back to C. This means he covers distance y to reach B, then another distance y to return to C, for a total distance of 2y. This entire second phase takes 30 minutes at the increased speed of 1.3v.
Now we can set up our equations using the distance equals velocity times time formula. For Phase 1, distance x equals velocity v times one-half hour, giving us x equals v over 2. For Phase 2, the total distance 2y equals velocity 1.3v times one-half hour, which gives us 2y equals 1.3v over 2. Simplifying the second equation, we get y equals 1.3v over 4.
Now we can find the final ratio. We have x equals v over 2 and y equals 1.3v over 4. To find the ratio x to y, we divide x by y. This gives us v over 2 divided by 1.3v over 4, which equals v over 2 times 4 over 1.3v. Simplifying, we get 4 over 2.6, which equals 20 over 13. Therefore, the ratio AC to BC is 20 to 13.