The boys were 1/
4 of the way across the bridge when they heard the sound of a train whistle. The train
coming from behind them was moving at a speed of 45 mph. The boys immediately thought that they could
turn around and run towards the near end of the bridge, and get to the end at the same time the train did. But,
they also knew that they could run to the far end of the bridge and get to the end at the same time the train
did.
• How fast were the boys running assuming they were 1/
4 of the way across the bridge?
• Suppose the boys were 2/
5 of the way across … how fast do they run?
• Suppose the boys were 1/
3 of the way across … how fast do they run?
• Suppose the boys were
3/
8 of the way across … how fast do they run?
•
• Write a generalization to solve this problem for any fractional amount.
To solve this problem, we can start by understanding the situation presented. The boys are on a bridge, and they hear a train coming towards them from behind. The boys have the option to either turn around and run towards the near end of the bridge or continue running towards the far end of the bridge. The goal is to determine how fast the boys need to run in order to reach the end of the bridge at the same time as the train.
Let's break down the problem by considering each scenario separately:
1. When the boys are 1/4 of the way across the bridge:
To determine the speed at which the boys need to run, we need to consider the relative speeds of the boys and the train. Since the train is coming from behind the boys, their effective speed is the difference between their individual speeds and the speed of the train. Let's suppose the boys' speed is v.
When the boys turn around and run towards the near end of the bridge, their effective speed will be v + 45 mph (train speed). When they run towards the far end of the bridge, their effective speed will be v - 45 mph.
Since they want to reach the end of the bridge at the same time as the train, the time it takes for them to run to the near end must equal the time it takes for them to run to the far end.
To determine their speed, we can set up an equation:
Distance to near end of the bridge / (v + 45) = Distance to far end of the bridge / (v - 45)
Since they are 1/4 of the way across the bridge, the distance to the near end is 1/4 of the total bridge length, and the distance to the far end is 3/4 of the total bridge length.
1/4 / (v + 45) = 3/4 / (v - 45)
Next, we can cross-multiply and solve for v:
4(v - 45) = 3(v + 45)
4v - 180 = 3v + 135
v = 315 mph
So, when the boys are 1/4 of the way across the bridge, they need to run at a speed of 315 mph to reach the end of the bridge at the same time as the train.
2. When the boys are 2/5 of the way across the bridge:
Using a similar approach, let's set up an equation to solve for their speed.
2/5 / (v + 45) = 3/5 / (v - 45)
Cross-multiplying and solving for v:
5(v - 45) = 3(v + 45)
5v - 225 = 3v + 135
v = 180 mph
So, when the boys are 2/5 of the way across the bridge, they need to run at a speed of 180 mph.
3. When the boys are 1/3 of the way across the bridge:
Setting up the equation:
1/3 / (v + 45) = 2/3 / (v - 45)
Solving for v:
3(v - 45) = 2(v + 45)
3v - 135 = 2v + 90
v = 225 mph
When the boys are 1/3 of the way across the bridge, they need to run at a speed of 225 mph.
4. When the boys are 3/8 of the way across the bridge:
Setting up the equation:
3/8 / (v + 45) = 5/8 / (v - 45)
Solving for v:
8(v - 45) = 5(v + 45)
8v - 360 = 5v + 225
v = 195 mph
When the boys are 3/8 of the way across the bridge, they need to run at a speed of 195 mph.
In general, to solve this problem for any fractional amount, you can follow these steps:
1. Let x represent the fractional distance the boys are across the bridge.
2. Set up the equation: x / (v + 45) = (1 - x) / (v - 45).
3. Cross-multiply and solve for v to find the required speed.
So, the generalization to solve this problem for any fractional amount is:
x / (v + 45) = (1 - x) / (v - 45), where x represents the fractional distance and v represents the boys' running speed.