Two runners at opposite ends of a 5 mile bridge begin jogging towards each other. Runner A is running at 5 mph, runner B is running at 4 mph. A mosquito lands on one runner as he starts on the bridge than proceeds to fly back and forth between the runners at 8 mph. How far has the mosquito flown when the runners meet.

To find how far the mosquito has flown when the runners meet, we need to determine the time it takes for the runners to meet.

Let's denote the distance from the starting point of Runner A to the point where the runners meet as x miles.

The distance covered by Runner A in this time is given by the equation:

Distance_A = Speed_A * Time = 5 mph * Time

Similarly, the distance covered by Runner B is given by the equation:

Distance_B = Speed_B * Time = 4 mph * Time

Since the mosquito flies back and forth between the runners at 8 mph, the distance covered by the mosquito is equal to twice the distance between them:

Distance_mosquito = 2 * x miles

Now, we can set up an equation to find the time it takes for the runners to meet:

Distance_A + Distance_B = 2 * x

5 mph * Time + 4 mph * Time = 2 * x

9 mph * Time = 2 * x

To find x, we need to determine the value of Time. We can use the formula:

Time = Distance / Speed

The distance is the total length of the bridge, which is 5 miles. Thus, we have:

Time = 5 miles / (5 mph + 4 mph)

Time = 5 miles / 9 mph

Now, we can substitute this value back into the equation to find x:

9 mph * (5 miles / 9 mph) = 2 * x

5 miles = 2 * x

Dividing both sides of the equation by 2, we get:

2.5 miles = x

Therefore, when the runners meet, the mosquito has flown a distance of:

Distance_mosquito = 2 * x = 2 * 2.5 miles = 5 miles

So, the mosquito has flown 5 miles when the runners meet.

To determine how far the mosquito has flown when the runners meet, we need to calculate the time it takes for the runners to meet first.

We can start by finding the time it takes for each runner to reach the meeting point. Let's call the distance Runner A travels as x miles. Since Runner A is running at 5 mph, the time taken by Runner A to reach the meeting point is given by:

Time taken by Runner A = Distance / Speed = x miles / 5 mph = (x/5) hours.

Since Runner B is running at 4 mph, the time taken by Runner B to reach the meeting point is given by:

Time taken by Runner B = Distance / Speed = [(5 - x) miles] / 4 mph = [(5 - x)/4] hours.

Now, since the mosquito is flying back and forth between the runners at 8 mph, its effective speed when the runners are moving towards each other is the sum of their speeds, which is 5 mph + 4 mph = 9 mph.

As the mosquito is moving back and forth between the runners, its total distance covered will be equal to the total time taken multiplied by its effective speed.

Total distance covered by the mosquito = Total time taken * Speed of the mosquito

Since the total time taken for the runners to meet is the same as the total time taken for the mosquito to fly back and forth, we can set up the equation:

Total distance covered by the mosquito = (x/5) hours * 9 mph = [(x/5) * 9] miles.

Therefore, the mosquito has flown [(x/5) * 9] miles when the runners meet.

To find the value of x (the distance Runner A travels), we can set up an equation using the relative speed of the runners:

Runner A's speed + Runner B's speed = Distance / Time

5 mph + 4 mph = 5 miles / Time

9 mph = 5 miles / Time

Time = 5 miles / 9 mph

Since Time = (x/5) hours (as calculated earlier), we can substitute this value back into the equation:

(x/5) hours = 5 miles / 9 mph

Cross-multiplying, we get:

x/5 = (5 * 1) / (9 * 1)

x/5 = 5/9

Cross-multiplying again, we get:

9x = 5 * 5

9x = 25

Dividing both sides by 9, we find:

x = 25/9

x ≈ 2.778 miles

Now we can substitute this value of x back into the total distance covered by the mosquito equation:

Total distance covered by the mosquito = [(x/5) * 9] miles

Total distance covered by the mosquito = [(2.778/5) * 9] miles

Total distance covered by the mosquito ≈ 5.0004 miles

Therefore, the mosquito has flown approximately 5.0004 miles when the runners meet.