A racing car driver must maintain an average speed of 180km/h over four rounds in order to qualify for a race. The racing csr completes the first round st an average speed of 170km/h and the second round at an average speed of 160km/h.
Calculate what the average speed over the last two rounds must be for the driver to qualify.
In order to calculate the average speed over the last two rounds, we need to first determine the total time it took to complete the first two rounds.
For the first round, the average speed was 170km/h. Let's assume the distance of the first round was D1. Then the time taken to complete the first round is given by:
Time1 = D1 / 170
Similarly, for the second round, the average speed was 160km/h. Let's assume the distance of the second round was D2. Then the time taken to complete the second round is given by:
Time2 = D2 / 160
To qualify for the race, the racing car driver must maintain an average speed of 180km/h over four rounds. Let's assume the distance of the last two rounds is D3. The average speed for the last two rounds is given by:
Average speed of last two rounds = (D3 + D4) / (Time3 + Time4)
Since the average speed is distance divided by time, we can re-write the equation as:
Average speed of last two rounds = (D3 + D4) / (D3 / 180 + D4 / 180)
We know that the average speed of the first round was 170 km/h and the average speed of the second round was 160 km/h. Therefore, we can set up two equations:
Average speed of first round = D1 / Time1 = 170
Average speed of second round = D2 / Time2 = 160
Rearranging these equations, we get:
D1 = 170 * Time1
D2 = 160 * Time2
Substituting these values into the equation for the average speed of the last two rounds, we have:
Average speed of last two rounds = (D3 + D4) / (D3 / 180 + D4 / 180)
= (D3 + D4) / (D3 * 180 / 170 + D4 * 180 / 160)
= (D3 + D4) / (180 * (D3 / 170) + 180 * (D4 / 160))
Since the total distance of the four rounds is fixed, we can express D4 in terms of D3 using the equation:
Total distance = D1 + D2 + D3 + D4
= 170 * Time1 + 160 * Time2 + D3 + D4
Rearranging this equation, we get:
D4 = Total distance - 170 * Time1 - 160 * Time2 - D3
Substituting D4 value in the equation for average speed of last two rounds, we have:
Average speed of last two rounds = (D3 + (Total distance - 170 * Time1 - 160 * Time2 - D3)) / (180 * (D3 / 170) + 180 * ((Total distance - 170 * Time1 - 160 * Time2 - D3) / 160))
= (Total distance - 170 * Time1 - 160 * Time2) / (180 * (D3 / 170) + 180 * ((Total distance - 170 * Time1 - 160 * Time2) / 160))
Given that the racing car driver must maintain an average speed of 180km/h over four rounds, we can set up another equation:
Total distance = Average speed * Total time
= 180 * (Time1 + Time2 + Time3 + Time4)
Substituting the values of Time1 and Time2 in the equation for total distance, we have:
Total distance = 180 * (D1 / 170 + D2 / 160 + Time3 + Time4)
Now, we can substitute the values of Time1, Time2, and the equation for Total distance into the equation for average speed of last two rounds:
Average speed of last two rounds = (Total distance - 170 * Time1 - 160 * Time2) / (180 * (D3 / 170) + 180 * ((Total distance - 170 * Time1 - 160 * Time2) / 160))
= ((180 * (D1 / 170 + D2 / 160 + Time3 + Time4)) - 170 * Time1 - 160 * Time2) / (180 * (D3 / 170) + 180 * ((180 * (D1 / 170 + D2 / 160 + Time3 + Time4)) - 170 * Time1 - 160 * Time2) / 160))
= ((180 * (D1 / 170 + D2 / 160 + Time3 + Time4)) - (170 * D1 / 170) - (160 * D2 / 160)) / ((180 * D3 / 170) + (180 * (D1 / 170 + D2 / 160 + Time3 + Time4)) - (170 * Time1) - (160 * Time2)))
Since the racing car driver must maintain an average speed of 180km/h over four rounds in order to qualify for the race, the average speed over the last two rounds is calculated by the above equation. This equation can be further simplified by substituting the given values for D1, D2, Time1, Time2, and Total distance.