Al and Bob, who live in North Vancouver, are Seattle Mariners fans. They regularly drive the 264km from their home to the ballpark in Seattle. On one particular day, Bob drove to the game. On the return journey, Al was able to increase their average speed by 10% and save 18 minutes on travelling time. Calculate the average speed at which Bob drove to the game.
Bob's speed --- x km/h
Al's speed ----- 1.1x km/h
Bob's time = 264/x and Al's time = 264/1.1x
the difference in their times is 18 min = 18/60 hours
264/x - 264/1.1x = 18/60 = 3/10
multiply each term by 110x , and take over to solve for x
comes out nice.
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.
To solve this problem, we can set up a system of equations. Let's denote Bob's average speed as "x" km/h.
We know that the distance traveled is 264 km. Therefore, the time Bob takes to drive to the game can be calculated as:
Time = Distance / Speed = 264 / x (equation 1)
Now, let's consider Al's return journey. We are given that Al increased their average speed by 10% and saved 18 minutes on traveling time.
Since Al increased their speed by 10%, their speed for the return journey would be 1.1 times Bob's speed, or 1.1x km/h.
The difference in time can be calculated as:
Time difference = Bob's time - Al's time = 18 minutes (equation 2)
To convert 18 minutes into hours, we divide by 60 since there are 60 minutes in an hour:
18 minutes / 60 = 0.3 hours
Now let's calculate Al's time using the increased speed:
Time = Distance / Speed = 264 / (1.1x) (equation 3)
Substituting equation 1 and equation 3 into equation 2, we get:
264 / x - 264 / (1.1x) = 0.3
To simplify the equation, we can multiply through by the least common multiple (LCM) of the denominators, which is 1.1x:
(1.1x) * (264 / x) - 264 = 0.3 * (1.1x)
1.1 * 264 - 264 = 0.33x
290.4 - 264 = 0.33x
26.4 = 0.33x
Now, we can solve for x by dividing both sides of the equation by 0.33:
26.4 / 0.33 = x
x ≈ 80 km/h
Therefore, Bob drove to the game at an average speed of approximately 80 km/h.