You hold a hose 45 degrees to the horizontal and at a height of 1 m from the floor. The water reaches a maximum distance of 10 m from where you are standing. Now you place your thumb over the end of the hose to occlude the opening by 80%, which in turn reduces the flow rate by 50%. Even though less fluid emerges the water travels further. How far does the water travel with your thumb over the end of the hose? Find the amount by which the hose must be occluded in order for the water to travel twice as far.

To find out how far the water travels when you place your thumb over the end of the hose, we can start by understanding how the flow rate affects the distance traveled.

1. Flow Rate and Distance: Generally, the distance traveled by water from a hose depends on the flow rate. Higher flow rates result in water traveling a shorter distance, while lower flow rates result in water traveling a longer distance. This is because higher flow rates generate more force which reduces the distance traveled.

In this scenario, when you occlude the opening of the hose by 80%, the flow rate reduces by 50%. This means that the water will now travel a longer distance compared to the initial scenario.

2. Calculating the new distance: To find the new distance the water travels when you occlude the hose by 80%, we need to calculate the ratio of the flow rates and use it to adjust the initial distance.

The flow rate is reduced by 50%, which means the new flow rate is 50% of the original flow rate. This forms a ratio of 50% (new flow rate) to 100% (original flow rate) or 1:2.

Since the new flow rate is half of the original, the water will travel twice as far. Therefore, the new distance will be 20 m (two times the initial distance of 10 m).

3. Finding the occlusion amount for double the distance: Now, let's determine the amount by which the hose must be occluded in order for the water to travel twice as far, which is 20 m.

Since occlusion reduces the flow rate, we can use the same ratio principle as before to find the occlusion amount. In this case, the ratio will be 20 m (double the initial distance) to 10 m (initial distance) or 2:1.

If we extend this ratio to the occlusion amount (percentage), we get 1 (occlusion amount) to 0.8 (initial occlusion amount) or 1:0.8.

To find the occlusion amount for double the distance, we can solve the following equation:

1 / x = 1 / 0.8

Cross multiplying, we get:

1 * 0.8 = 1 * x

0.8 = x

Therefore, the hose must be occluded by 80% (0.8) in order for the water to travel twice as far.

Overall, when you occlude the hose by 80%, the water will travel twice as far, and the occlusion amount required for the water to travel twice as far is also 80% (0.8).