Mathematics, concrete

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Anu Bhardwaj · Accepted Answer

The answer I provided is based on an exponential model fitted using just two points from your data set. While this provides an estimate, it's important to note that fitting a model with more data points, or using different methods, could indeed yield different results. To fit an exponential equation manually, we'll first need to hypothesize the general form of the equation, which for an exponential relationship is typically: 𝑄 = 𝑎⋅𝑒^(𝑏⋅ℎ) Where: 𝑄 is the discharge in Cumecs. ℎ is the gauge reading in meters. 𝑎 and 𝑏 are constants we need to determine. From the data provided, we choose two points to create two equations and solve for 𝑎 and 𝑏 . Let's choose two points from the table: Point 1: ℎ =0.63 m, 𝑄 =2228.6 Cumecs Point 2: ℎ =5.43 m, 𝑄 =2418.2 Cumecs We set up the equations: 2228.6= 𝑎⋅𝑒^(𝑏⋅ 0.63) 2418.2= 𝑎⋅𝑒^(𝑏⋅ 5.43) To solve for 𝑏 , we can divide the second equation by the first: 2418.2/2228.6= 𝑎⋅𝑒^(𝑏⋅ 5.43)/ 𝑎⋅𝑒^(𝑏⋅ 0.63)            2418.2/2228.6= 𝑒^𝑏⋅ (5.43−0.63)         Let's calculate 𝑏 b: 𝑒^(𝑏⋅ 4.8) =2418.2/2228.6     𝑏⋅ 4.8=ln (2418.2/2228.6) 𝑏 =ln (2418.2/2228.6)/4.8 We can then find 𝑎 a by plugging 𝑏 b back into one of the original equations. Let’s perform the calculations. From our calculations, the constants are approximately 𝑎 ≈2204.84a≈2204.84 and 𝑏 ≈0.017b≈0.017. Therefore, the fitted exponential equation is: 𝑄 =2204.84 ⋅𝑒^ 0.017 ⋅ℎ Now, let's use this equation to find the discharges for gauge readings of 104 m and 187 m. ​​ The calculated discharges are: For a staff gauge reading of 104 m, the discharge is 12918.43Cumecs. For a staff gauge reading of 187 m, the discharge is 52966.251Cumecs you can choose any pair of points from the table to fit the exponential model, or even better, use multiple points to establish a more robust model. When selecting points, consider these strategies to potentially improve the accuracy and reliability of the fit: Spread of Data: Choose points that are spread across the entire range of the dataset rather than clustered in one area. This helps in capturing the overall trend more effectively. Representative Points: Select points that you believe are representative of the underlying process or trend. Avoid points that might be outliers or anomalies unless you are sure of their accuracy and relevance. Multiple Points : Instead of just two points, using multiple points (three or more) can help in fitting a more accurate model. This is particularly useful when the relationship might not be strictly exponential, or when the data shows some variability. Error Minimization: In a more sophisticated approach, you can use all points and apply a method like least squares to minimize the error across all predictions, not just at one or two points. This generally requires using statistical software or a programming approach. By taking into account more data points and considering their distribution, you can derive a model that better represents the entire dataset, leading to more reliable predictions.

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