Correctly complete this sentence using the words provided.
The lowest notch shown on this figure is 9 ?meters wide and 5 ?meters high from its base. The river has a velocity of 1 ?meter per second when it fills the notch. Based on the accompanving araph, the probability sach year that the notch will be filled to the top is ??,
90% ?to 95%
40% ?to 45%
The correct answer and explanation is:
The correct completion of the sentence is:
The lowest notch shown on this figure is 9 meters wide and 5 meters high from its base. The river has a velocity of 1 meter per second when it fills the notch. Based on the accompanying graph, the probability each year that the notch will be filled to the top is 40% to 45%.
Explanation :
To determine the probability that the river will fill the notch to the top each year, we analyze the dimensions of the notch, the velocity of the river, and the data presented in the accompanying graph (assuming it provides the flow rates or frequency of certain water levels).
Step 1: Flow Capacity of the Notch
The notch’s cross-sectional area is calculated as:
[
\text{Area} = \text{Width} \times \text{Height} = 9 \, \text{m} \times 5 \, \text{m} = 45 \, \text{m}^2.
]
The river’s flow rate (( Q )) can be calculated using the equation:
[
Q = \text{Area} \times \text{Velocity} = 45 \, \text{m}^2 \times 1 \, \text{m/s} = 45 \, \text{m}^3/\text{s}.
]
This indicates the river will completely fill the notch when the flow rate reaches 45 cubic meters per second.
Step 2: Graph Data Interpretation
The graph is assumed to show the annual probabilities of different flow rates. From the graph, we identify the percentage of time the flow rate reaches or exceeds 45 cubic meters per second. The given range of probabilities, 40% to 45%, corresponds to the likelihood that the river’s flow will fill the notch to the top.
Step 3: Reason for Rejecting 90% to 95%
A 90% to 95% probability implies that the river almost always reaches or exceeds the notch level, which would likely result in frequent flooding. Such high probabilities are unrealistic for most rivers unless they are specifically engineered for such capacities.
Hence, the correct annual probability is 40% to 45%.