1. **State the problem:** Two dogs start 500 feet apart and run toward each other at 10 ft/sec each. A flea flies back and forth between them at 25 ft/sec until the dogs collide. We need to find the total distance the flea flies. 2. **Understand the scenario:** The dogs are moving toward each other, so their relative speed is the sum of their speeds: $$10 + 10 = 20\ \text{ft/sec}$$. 3. **Calculate the time until collision:** Since they start 500 feet apart and close the gap at 20 ft/sec, the time until collision is: $$t = \frac{\text{distance}}{\text{relative speed}} = \frac{500}{20} = 25\ \text{seconds}$$ 4. **Calculate the distance the flea flies:** The flea flies continuously at 25 ft/sec for the entire 25 seconds until the dogs collide, so: $$\text{distance flown by flea} = \text{speed} \times \text{time} = 25 \times 25 = 625\ \text{feet}$$ **Final answer:** The flea flies a total of 625 feet.