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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Let's tackle question 4, which involves analyzing tile designs. a) Draw the fourth design. Step 1: Analyze the pattern. Pattern 1: A 1 × 1 red tile surrounded by white tiles, forming a 3 × 3 square. Pattern 2: A 2 × 2 red tile square surrounded by white tiles, forming a 4 × 4 square. Pattern 3: A 3 × 3 red tile square surrounded by white tiles, forming a 5 × 5 square. The pattern shows that for pattern n, there is an n × n square of red tiles, and the total square (red and white tiles) is (n+2) × (n+2). Step 2: Draw the fourth design. For Pattern 4, there will be a 4 × 4 square of red tiles, surrounded by white tiles, forming a 6 × 6 square in total. ` ⬜⬜⬜⬜⬜⬜ ⬜🟥🟥🟥🟥⬜ ⬜🟥🟥🟥🟥⬜ ⬜🟥🟥🟥🟥⬜ ⬜🟥🟥🟥🟥⬜ ⬜⬜⬜⬜⬜⬜ ` (Here, ⬜ represents a white tile and 🟥 represents a red tile.) b) Complete the given table to illustrate what happens to the number of white tiles when compared to the red tiles. Step 1: Count red and white tiles for Patterns 1, 2, and 3. For Pattern n: Number of red tiles = n^2 Total tiles = (n+2)^2 Number of white tiles = Total tiles - Number of red tiles = (n+2)^2 - n^2 Pattern 1: Red tiles = 1^2 = 1 White tiles = (1+2)^2 - 1^2 = 3^2 - 1 = 9 - 1 = 8 Pattern 2: Red tiles = 2^2 = 4 White tiles = (2+2)^2 - 2^2 = 4^2 - 4 = 16 - 4 = 12 Pattern 3: Red tiles = 3^2 = 9 White tiles = (3+2)^2 - 3^2 = 5^2 - 9 = 25 - 9 = 16 Step 2: Calculate red and white tiles for Patterns 4, 5, and 10. Pattern 4: Red tiles = 4^2 = 16 White tiles = (4+2)^2 - 4^2 = 6^2 - 16 = 36 - 16 = 20 Pattern 5: Red tiles = 5^2 = 25 White tiles = (5+2)^2 - 5^2 = 7^2 - 25 = 49 - 25 = 24 Pattern 10: Red tiles = 10^2 = 100 White tiles = (10+2)^2 - 10^2 = 12^2 - 100 = 144 - 100 = 44 Step 3: Complete the table. |l|c|c|c|c|c|c| Pattern & 1 & 2 & 3 & 4 & 5 & 10 \\ Red tiles & 1 & 4 & 9 & 16 & 25 & 100 \\ White tiles & 8 & 12 & 16 & 20 & 24 & 44 \\ c) Write down the rule(s) you used to extend the pattern. Step 1: Describe the rule for red tiles. The number of red tiles in pattern n is n squared. Rule for Red Tiles: Red tiles = n^2. Step 2: Describe the rule for white tiles. The number of white tiles increases by 4 for each subsequent pattern. Alternatively, the number of white tiles in pattern n can be found using the formula: White tiles = (n+2)^2 - n^2 = (n^2 + 4n + 4) - n^2 = 4n + 4. Rule for White Tiles: White tiles = 4n + 4. d) Provide a rule to describe the relationship between the following numbers, by making use of number sentences. 9; 14; 19; 24; ... and determine the 20th term. Step 1: Find the common difference. The sequence is 9, 14, 19, 24, ... The difference between consecutive terms is: 14 - 9 = 5 19 - 14 = 5 24 - 19 = 5 The common difference is 5. This is an arithmetic sequence. Step 2: Write the rule for the n-th term. The formula for the n-th term of an arithmetic sequence is T_n = a + (n-1)d, where a is the first term and d is the common difference. Here, a = 9 and d = 5. T_n = 9 + (n-1)5 T_n = 9 + 5n - 5 T_n = 5n + 4 The rule is T_n = 5n + 4. Step 3: Determine the 20th term. Substitute n=20 into the rule: T_20 = 5(20) + 4 T_20 = 100 + 4 T_20 = 104 The 20th term is 104. What's next? Send 'em! 📸