Here's how you can approach solving this network adjustment problem using Excel as requested: --- ### **Step 1: Input Data in Excel** #### **Coordinates and Azimuth** - Q = (1000.00, 1000.00) - Azimuth QR = 0°06'24.5" = 0.1068° = 0.001864 radians #### **Distance Observations Table** Create columns for: - From - To - Distance (ft) - Std. Dev (S ft) #### **Angle Observations Table** Create columns for: - Back sight - Instrument - Fore sight - Angle (deg, min, sec) - Std. Dev (S ") #### **Azimuth Observation Table** Create columns for: - From - To - Azimuth (deg, min, sec) - Std. Dev (S ") --- ### **Step 2: Approximate Coordinates** Input the initial approximations for: - R: (1003.07, 2640.00) - S: (2323.07, 2638.46) - T: (2661.74, 1096.08) --- ### **Step 3: Set Up Observation Equations** #### **Distance Equation** For distance between (xi, yi) and (xj, yj): \[ l_{ij} = \sqrt{(x_j - x_i)^2 + (y_j - y_i)^2} \] #### **Angle Equation** For angle at station i between points j and k: \[ \theta_{jik} = \arctan\left(\frac{y_k - y_i}{x_k - x_i}\right) - \arctan\left(\frac{y_j - y_i}{x_j - x_i}\right) \] #### **Azimuth Equation** \[ Az_{ij} = \arctan\left(\frac{x_j - x_i}{y_j - y_i}\right) \] --- ### **Step 4: Linearize the Equations** Use Taylor series expansions to linearize the equations around your initial approximations. Use partial derivatives to relate corrections in coordinates to corrections in observed values. --- ### **Step 5: Formulate Least Squares Adjustment (in Excel)** 1. **Design Matrix (A):** Use partial derivatives as coefficients for corrections to X and Y of unknowns. 2. **Observation Vector (L):** Differences between observed and computed values. 3. **Weight Matrix (P):** Inverse of variances (from given standard deviations). 4. **Normal Equation:** \[ (A^T P A) \Delta X = A^T P L \] 5. **Solve for Corrections (\(\Delta X\)):** Use Excel’s matrix functions (e.g., `MMULT`, `MINVERSE`, `TRANSPOSE`). --- ### **Step 6: Iteratively Update Coordinates** - Update the approximated coordinates with the computed corrections. - Recompute the observation equations with updated coordinates. - Repeat until corrections are sufficiently small. --- ### **Step 7: Compute Residuals** \[ v = L - A \Delta X \] --- ### **Step 8: Output Results** - Final coordinates for R, S, and T - Residuals for each observation - Standard deviations (from the covariance matrix, \( (A^T P A)^{-1} \)) - Adjusted observations --- ## **Excel Implementation Tips** - **Use separate sheets** for observations, calculations, and results. - **Partial derivatives**: Calculate these in helper cells for each equation. - **Matrix operations**: Use named ranges and `MMULT`, `MINVERSE` to solve normal equations. --- ## **What to Submit** 1. Final adjusted coordinates for R, S, T. 2. Table of residuals for each observation (distance, angle, azimuth). 3. Standard deviations of adjusted coordinates. 4. Standard deviations of adjusted observations. --- ### **If you need a sample Excel structure or formulas for a specific part, let me know! I can provide an example for setting up matrices or the partial derivatives.** --- **Summary Table Example (for submission):** | Point | X (ft) | Y (ft) | Std. Dev X | Std. Dev Y | |-------|----------|----------|------------|------------| | R | xxxx.xx | xxxx.xx | xx.xx | xx.xx | | S | xxxx.xx | xxxx.xx | xx.xx | xx.xx | | T | xxxx.xx | xxxx.xx | xx.xx | xx.xx | | Observation | Residual | |-------------|----------| | Q-R | xx.xx | | ... | ... | --- **Let me know if you want a sample Excel file template or step-by-step formulas!**