1. **Problem Statement:** Two cars, A and B, start from rest and accelerate. Given velocity-time graphs for both cars with $a=3$ minutes, we need to determine which car is ahead after 3 and 6 minutes, interpret the shaded area between the curves, and estimate when the cars are side by side again. 2. **Key Concept:** The distance traveled by a car is the area under its velocity-time graph. The difference in distance between two cars at a given time is the area between their velocity curves. 3. **(a) Which car is ahead after 3 minutes?** - Since $a=3$, after 3 minutes corresponds to time $t=a$. - The area under curve A from 0 to $a$ is greater than the area under curve B. - Therefore, car A has traveled a greater distance and is ahead after 3 minutes. 4. **(b) Meaning of the shaded area:** - The shaded region between curves A and B from 0 to $a$ represents the difference in distance traveled by the two cars. - Since car A is ahead, the shaded area equals the distance by which A is ahead of B after 3 minutes. 5. **(c) Which car is ahead after 6 minutes?** - At $t=2a=6$ minutes, the area under curve A is still greater than under curve B. - Hence, car A remains ahead after 6 minutes. 6. **(d) Estimate time $t$ when cars are side by side again:** - Cars are side by side when the net difference in distance traveled is zero. - This occurs when the total area between the curves from 0 to $t$ is zero. - From the graph, the shaded area between $a$ and $2a$ is below curve A, indicating car B gains on A. - The time $t$ when areas above and below balance is approximately $t=4.5$ minutes. **Final answers:** - (a) Car A is ahead after 3 minutes. - (b) The shaded area is the distance by which A is ahead of B after 3 minutes. - (c) Car A is ahead after 6 minutes. - (d) Cars are side by side again at approximately $t=4.5$ minutes.