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The following rectangular figures are made of toothpicks. a. How many toothpicks are needed to build the fourth figure? b. How many toothpicks are needed to build the 20th figure? 1st 2d 3d

The following rectangular figures are made of toothpicks. a. How many toothpicks are needed to build the fourth figure? b. How many toothpicks are needed to build the 20th figure? 1st 2d 3d

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Rectangular Figures Made of Toothpicks** The following rectangular figures are made of toothpicks. **Question a:** How many toothpicks are needed to build the fourth figure? **Question b:** How many toothpicks are needed to build the 20th figure? **Explanation and Diagrams:** Below are diagrams representing the sequential addition of rectangles made from toothpicks: 1. **1st figure**: Two rectangles are formed, labelled as "1st." - Diagram: A single rectangle made with 4 toothpicks. 2. **2nd figure**: Additional rectangles are added, forming four rectangles, labelled as "2d." - Diagram: A series of two rectangles horizontally concatenated, sharing a common side, resulting in a total of 7 toothpicks. 3. **3rd figure**: More rectangles are incorporated, forming six rectangles, labelled as "3d." - Diagram: A series of three rectangles horizontally concatenated, sharing common sides, resulting in a total of 10 toothpicks. **Degree of Increase:** - Observing the pattern, each additional figure adds 3 more toothpicks. This can be deduced from the given increments: - 1st figure: 4 toothpicks - 2nd figure: 7 toothpicks (4 + 3) - 3rd figure: 10 toothpicks (7 + 3) **Answer Section:** a. **The answer is** ___13___. - Fourth figure: 10 + 3 = 13 toothpicks b. **The answer is** ___61___. - 20th figure: Follows the same pattern. Starting with 4 toothpicks, each figure adds 3 toothpicks - Using the formula for the nth term in an arithmetic sequence: a + (n - 1)d - Where "a" is the initial number of toothpicks (4), - "d" is the common difference (3 toothpicks), - "n" is the figure number (20th figure). Thus, T_20 = 4 + (20 - 1) * 3 = 4 + 57 = 61 toothpicks.
Transcribed Image Text:**Rectangular Figures Made of Toothpicks** The following rectangular figures are made of toothpicks. **Question a:** How many toothpicks are needed to build the fourth figure? **Question b:** How many toothpicks are needed to build the 20th figure? **Explanation and Diagrams:** Below are diagrams representing the sequential addition of rectangles made from toothpicks: 1. **1st figure**: Two rectangles are formed, labelled as "1st." - Diagram: A single rectangle made with 4 toothpicks. 2. **2nd figure**: Additional rectangles are added, forming four rectangles, labelled as "2d." - Diagram: A series of two rectangles horizontally concatenated, sharing a common side, resulting in a total of 7 toothpicks. 3. **3rd figure**: More rectangles are incorporated, forming six rectangles, labelled as "3d." - Diagram: A series of three rectangles horizontally concatenated, sharing common sides, resulting in a total of 10 toothpicks. **Degree of Increase:** - Observing the pattern, each additional figure adds 3 more toothpicks. This can be deduced from the given increments: - 1st figure: 4 toothpicks - 2nd figure: 7 toothpicks (4 + 3) - 3rd figure: 10 toothpicks (7 + 3) **Answer Section:** a. **The answer is** ___13___. - Fourth figure: 10 + 3 = 13 toothpicks b. **The answer is** ___61___. - 20th figure: Follows the same pattern. Starting with 4 toothpicks, each figure adds 3 toothpicks - Using the formula for the nth term in an arithmetic sequence: a + (n - 1)d - Where "a" is the initial number of toothpicks (4), - "d" is the common difference (3 toothpicks), - "n" is the figure number (20th figure). Thus, T_20 = 4 + (20 - 1) * 3 = 4 + 57 = 61 toothpicks.
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