Train A starts at 4 miles South of a bridge and heads North at a constant speed of 30 miles per hour. Train B starts 6 miles North of the bridge.

a. What velocity must Train B have so that the two trains cross the bridge at the same time?

b. If Train B goes at 35 miles per hour, South, how far away from the bridge do they cross?

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To answer these questions, let's start by understanding the distances and speeds involved:

- Train A starts 4 miles South of the bridge
- Train B starts 6 miles North of the bridge
- Train A travels North at a constant speed of 30 miles per hour
- Train B's velocity is unknown (part a) or 35 miles per hour South (part b)

a. To determine the velocity Train B must have so that the two trains cross the bridge at the same time, we need to compare the distances both trains have to cover.

Train A needs to cover a distance of 4 miles (to reach the bridge), while Train B needs to cover a distance of 6 miles (to reach the bridge in the opposite direction).

Since both trains are traveling towards each other, their combined distance should remain the same. Therefore, Train A (traveling at 30 miles per hour) will need to cover the distance of 4 miles, while Train B (traveling at an unknown velocity, let's call it v) will need to cover the distance of 6 miles.

Using the formula: distance = speed x time, we can set up the equation:

Distance covered by Train A = Distance covered by Train B
30 miles/hour x time for Train A = v miles/hour x time for Train B

Since time is the same for both trains (as they will cross the bridge at the same time), we can simplify the equation to:

30 miles/hour x time = v miles/hour x time

The time can be canceled out, leaving us with:

30 miles/hour = v miles/hour

Therefore, Train B must have a velocity of 30 miles per hour in the opposite direction (South) to cross the bridge at the same time as Train A.

b. If Train B goes at 35 miles per hour, South, we can calculate the distance at which they cross.

Since Train A is still traveling at 30 miles per hour, and Train B is traveling in the opposite direction at 35 miles per hour, their combined speed is:

Combined speed = Speed of Train A + Speed of Train B = 30 miles/hour + 35 miles/hour = 65 miles/hour

To calculate the distance at which they cross, we can use the formula: distance = speed x time.

Since the combined speed is 65 miles/hour, and they both start at different distances from the bridge, we need to find out how long it would take for them to meet. Let's denote this time as t.

For Train A:
Distance covered by Train A = Speed x Time
Distance covered by Train A = 30 miles/hour x t

For Train B:
Distance covered by Train B = Speed x Time
Distance covered by Train B = 35 miles/hour x t

To find the distance at which they cross, we need to equate the two distances:

30 miles/hour x t = 35 miles/hour x t

Since the time is the same for both trains (as they will cross the bridge at the same time), we can simplify the equation to solve for t:

30 miles/hour = 35 miles/hour

This is not possible since 30 miles/hour does not equal 35 miles/hour. It means that the two trains will not cross the bridge at the same time if Train B is traveling at 35 miles per hour in the opposite direction (South).

Therefore, for part b, we cannot determine the exact distance at which they cross using the given information.

To find the answers to these questions, we can calculate the time it takes for each train to reach the bridge.

First, let's find the time it takes for Train A to reach the bridge:
Distance traveled by Train A = Distance from starting point to the bridge = 4 miles
Speed of Train A = 30 miles per hour

Using the formula time = distance / speed, we can calculate the time (t) it takes for Train A to reach the bridge:
t = 4 miles / 30 miles per hour.

a. To find the velocity at which Train B needs to travel so that the trains cross the bridge at the same time, we need to consider the distance traveled by Train B. Since Train B starts 6 miles North of the bridge, the distance traveled by Train B when it crosses the bridge is 6 miles as well.

Let's assume the velocity of Train B is v miles per hour.
Hence, the time it takes for Train B to reach the bridge can be calculated as:
t = 6 miles / v miles per hour.

Since both trains are crossing the bridge at the same time, the time for Train A and Train B should be equal. Therefore, we can equate the time expressions for Train A and Train B:
4 miles / 30 miles per hour = 6 miles / v miles per hour.

Cross-multiplying and solving for v, we get:
4v = 6 * 30,
4v = 180,
v = 180 / 4,
v = 45 miles per hour.

Therefore, Train B needs to travel at a velocity of 45 miles per hour to cross the bridge at the same time as Train A.

b. If Train B goes at 35 miles per hour, South, we can find out how far away from the bridge the trains cross.

Knowing the velocity of Train B, we can calculate the time it takes for Train B to reach the bridge:
t = 6 miles / 35 miles per hour.

The distance traveled by Train A by the time they cross is:
Distance traveled by Train A = Speed of Train A * Time taken by Train B to reach the bridge.

Hence, the distance traveled by Train A can be calculated as:
Distance traveled by Train A = 30 miles per hour * (6 miles / 35 miles per hour).

Simplifying the expression, the distance traveled by Train A is:
Distance traveled by Train A = (30 * 6) / 35 miles.

Therefore, the trains cross a distance of:
Distance from the bridge = Distance traveled by Train A + 4 miles.

Substituting the values, the distance from the bridge where they cross is:
Distance from the bridge = ((30 * 6) / 35) + 4 miles.

Answer