Let's assume that Bob's average speed on the drive to the game was S km/h.
We know that the distance from Al and Bob's home to the ballpark is 264 km.
The time it took Bob to drive to the game can be calculated using the formula:
Time = Distance / Speed
So, the time taken by Bob to drive to the game was:
Time taken by Bob = 264 km / S km/h
Now let's calculate the time Al took on the return journey.
Al was able to increase their average speed by 10%, so his speed on the return journey was 1.10S km/h.
Saving 18 minutes on travelling time means that Al took 18 minutes less than Bob. Since both Al and Bob drove the same distance, we can set up the equation:
Time taken by Bob - 18 minutes = Time taken by Al
Converting 18 minutes to hours, we get 18 minutes / 60 minutes/hour = 0.3 hours.
So, the equation becomes:
264 km / S km/h - 0.3 hours = 264 km / (1.10S) km/h
Now, let's solve the equation to find the value of S.
Multiplying both sides of the equation by S and 1.10S to remove the denominators, we get:
(264 km / S km/h - 0.3 hours) * S * 1.10S = 264 km
Simplifying:
(264 - 0.3S) * 1.10S = 264
Expanding:
290.4S - 0.33S^2 = 264
Rearranging the equation:
0.33S^2 - 290.4S + 264 = 0
Now we can solve this quadratic equation using the quadratic formula:
S = (-b ± √(b^2 - 4ac)) / 2a
Where a = 0.33, b = -290.4, and c = 264.
Calculating:
S = (-(-290.4) ± √((-290.4)^2 - 4(0.33)(264))) / (2 * 0.33)
Simplifying:
S = (290.4 ± √(84456.96 - 348.48)) / 0.66
S = (290.4 ± √84108.48) / 0.66
S = (290.4 ± 290.0) / 0.66
We have two possible values for S, one positive and one negative, but since we're looking for a positive average speed, we take the positive value:
S = (290.4 + 290.0) / 0.66
S = 580.4 / 0.66
S ≈ 878.79 km/h
Therefore, the average speed at which Bob drove to the game was approximately 878.79 km/h.