This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for the given problems. Part (c): Complete the magic square The problem asks to complete the magic square such that the sum of numbers in each row, column, and diagonal is 18. The given partial square is: x & x & 4 \\ x & x & x \\ 7 & 8 & x Let the square be represented by variables: M_11 & M_12 & M_13 \\ M_21 & M_22 & M_23 \\ M_31 & M_32 & M_33 From the image, we have M_13=4, M_31=7, M_32=8. The magic sum is S=18. Step 1: Find M_33 using the sum of the third row. M_31 + M_32 + M_33 = 18 7 + 8 + M_33 = 18 15 + M_33 = 18 M_33 = 18 - 15 M_33 = 3 The square becomes: x & x & 4 \\ x & x & x \\ 7 & 8 & 3 Step 2: Find M_22 using the sum of the anti-diagonal (M_13, M_22, M_31). M_13 + M_22 + M_31 = 18 4 + M_22 + 7 = 18 11 + M_22 = 18 M_22 = 18 - 11 M_22 = 7 The square becomes: x & x & 4 \\ x & 7 & x \\ 7 & 8 & 3 Step 3: Find M_11 using the sum of the main diagonal (M_11, M_22, M_33). M_11 + M_22 + M_33 = 18 M_11 + 7 + 3 = 18 M_11 + 10 = 18 M_11 = 18 - 10 M_11 = 8 The square becomes: 8 & x & 4 \\ x & 7 & x \\ 7 & 8 & 3 Step 4: Find M_12 using the sum of the first row. M_11 + M_12 + M_13 = 18 8 + M_12 + 4 = 18 12 + M_12 = 18 M_12 = 18 - 12 M_12 = 6 The square becomes: 8 & 6 & 4 \\ x & 7 & x \\ 7 & 8 & 3 Step 5: Find M_21 using the sum of the first column. M_11 + M_21 + M_31 = 18 8 + M_21 + 7 = 18 15 + M_21 = 18 M_21 = 18 - 15 M_21 = 3 The square becomes: 8 & 6 & 4 \\ 3 & 7 & x \\ 7 & 8 & 3 Step 6: Find M_23 using the sum of the second row. M_21 + M_22 + M_23 = 18 3 + 7 + M_23 = 18 10 + M_23 = 18 M_23 = 18 - 10 M_23 = 8 The completed square is: 8 & 6 & 4 \\ 3 & 7 & 8 \\ 7 & 8 & 3 Let's verify all sums: Rows: 8+6+4 = 18 (Correct) 3+7+8 = 18 (Correct) 7+8+3 = 18 (Correct) Columns: 8+3+7 = 18 (Correct) 6+7+8 = 21 (Incorrect, should be 18) 4+8+3 = 15 (Incorrect, should be 18) The given numbers (4, 7, 8) are inconsistent with the properties of a 3x3 magic square where all rows, columns, and diagonals must sum to 18. Specifically, the center element of a 3x3 magic square must be one-third of the magic sum (18/3 = 6). However, based on the given numbers, the anti-diagonal sum forces the center element to be 7. This contradiction means a complete magic square cannot be formed with the given numbers and sum. Since a complete magic square cannot be formed, we provide the square filled as much as possible following the initial steps, and note the inconsistency. The completed square based on the initial steps is: 8 & 6 & 4 \\ 3 & 7 & 8 \\ 7 & 8 & 3 Note: This square satisfies the sum of all rows and both diagonals, but not all columns. Part (d): Find the value of s The given equation is V^2 - U^2 = 2as. We are given the values: V=3, U=2, a=1. We need to find the value of s. Step 1: Substitute the given values into the equation. (3)^2 - (2)^2 = 2(1)s Step 2: Calculate the squares. 9 - 4 = 2s Step 3: Simplify the left side. 5 = 2s Step 4: Solve for s. s = (5)/(2) s = 2.5 The value of s is 2.5. Send me the next one 📸