2026 ACJC17 C1BT1 Q8

Brand \(X\) and Brand \(Y\) batteries were tested to find out which brand has a longer battery life. For the test, a toy car was released from a starting point and was allowed to run continuously on a circular racing track until it stopped on its own. Each lap of the circular racing track is \(20\) metres.

1. When the toy car used Brand \(X\) battery, it completed the first lap in \(15\) seconds and each subsequent lap took \(3\) seconds longer than the previous lap.
   1. In which lap did the toy car first have a lap timing of more than \(312.3\) seconds.[2]
   2. Given that the toy car ran for \(23\) hours before stopping, find the average speed of the car during the last incomplete lap.[4]
2. When the toy car used Brand \(Y\) battery, it completed the first lap in \(7.68\) seconds and the time taken for each subsequent lap was \(2.5\%\) more than the time for the previous lap. Given that the toy car stopped upon completing the \(230\text{th}\) lap, determine which brand of battery has a longer battery life. You must show sufficient working to justify your answer.[2]

In another test using Brand \(Y\) battery, the toy car was made to stop for \(5\) seconds each time it completed one lap of the circular racing track. The time taken for the toy car to complete the first lap was still \(7.68\) seconds. However, the time taken for the toy car to complete each subsequent lap was \(2.5\%\) more than the total time taken for the previous lap and the \(5\) seconds stop time. For example the time taken for the toy car to complete the second lap was \(1.025 \times (7.68 + 5)\) seconds.

3. Show that the time taken for the toy car to complete the \(n\text{th}\) lap is \((212.68)(1.025)^{n-1} - 205\) seconds. Hence find the number of laps whereby the time taken to complete a lap is less than one hour.[4]