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Sagot :
To solve the problem, let's complete the table step-by-step and analyze the number of shaded triangles and their areas as they progress through each step.
### Step-by-Step Solution:
#### Step 0:
- Number of shaded triangles: 1 (initial triangle)
- Area of each shaded triangle: 256 square inches
#### Step 1:
- Number of shaded triangles:
- In Step 1, the initial triangle is divided into 4 smaller congruent equilateral triangles, and the middle one is removed.
- Thus, the number of shaded triangles = 3.
- Area of each shaded triangle:
- Each of the 3 remaining triangles has 1/4 of the original area.
- Area = 256 / 4 = 64 square inches
#### Step 2:
- Number of shaded triangles:
- In Step 2, each of the 3 smaller triangles from Step 1 is divided again into 4 smaller equilateral triangles, removing the middle one from each group.
- So, the number of shaded triangles is 3 (previous step) 3 = 9.
- Area of each shaded triangle:
- Each of the new triangles has 1/4 of the area of the triangles from Step 1.
- Area = 64 / 4 = 16 square inches
#### Step 3:
- Number of shaded triangles:
- In Step 3, each of the 9 triangles from Step 2 is again divided and the middle triangle removed.
- Number of shaded triangles = 9 (previous step) 3 = 27.
- Area of each shaded triangle:
- Each of the new triangles now has 1/4 of the area of the triangles from Step 2.
- Area = 16 / 4 = 4 square inches
#### Step 4:
- Number of shaded triangles:
- In Step 4, the same process is repeated for each of the 27 triangles from Step 3.
- Number of shaded triangles = 27 (previous step) 3 = 81.
- Area of each shaded triangle:
- Each of the new triangles has 1/4 of the area of the triangles from Step 3.
- Area = 4 / 4 = 1 square inch
#### Step 5:
- Number of shaded triangles:
- In Step 5, repeat the process once more for each of the 81 triangles from Step 4.
- Number of shaded triangles = 81 (previous step) 3 = 243.
- Area of each shaded triangle:
- Each of the new triangles has 1/4 of the area of the triangles from Step 4.
- Area = 1 / 4 = 0.25 square inches
### Completed Table:
[tex]\[ \begin{array}{|l|l|l|} \hline \text{Step \#} & \begin{array}{l} \# \\ \text{of} \\ \text{shaded} \\ \text{triangle} \end{array} & \begin{array}{l} \text{Area of} \\ \text{each} \\ \text{shaded} \\ \text{triangle} \\ (\text{square inches}) \end{array} \\ \hline 0 & 1 & 256 \\ \hline 1 & 3 & 64.0 \\ \hline 2 & 9 & 16.0 \\ \hline 3 & 27 & 4.0 \\ \hline 4 & 81 & 1.0 \\ \hline 5 & 243 & 0.25 \\ \hline \end{array} \][/tex]
### Graphs:
1. Number of Shaded Triangles as a Function of the Step Number:
[tex]\[ \begin{array}{|c|c|} \hline \text{Step \#} & \# \text{ of Shaded Triangles} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline 4 & 81 \\ \hline 5 & 243 \\ \hline \end{array} \][/tex]
Plot these points on a graph with the step number on the x-axis and the number of shaded triangles on the y-axis. The pattern shows exponential growth, specifically [tex]\(3^n\)[/tex] where [tex]\(n\)[/tex] is the step number.
2. Area of Each Shaded Triangle as a Function of the Step Number:
[tex]\[ \begin{array}{|c|c|} \hline \text{Step \#} & \text{Area (square inches)} \\ \hline 0 & 256 \\ \hline 1 & 64.0 \\ \hline 2 & 16.0 \\ \hline 3 & 4.0 \\ \hline 4 & 1.0 \\ \hline 5 & 0.25 \\ \hline \end{array} \][/tex]
Plot these points on a graph with the step number on the x-axis and the area of each shaded triangle on the y-axis. This pattern shows exponential decay, specifically [tex]\(\frac{256}{4^n}\)[/tex] where [tex]\(n\)[/tex] is the step number.
### Step-by-Step Solution:
#### Step 0:
- Number of shaded triangles: 1 (initial triangle)
- Area of each shaded triangle: 256 square inches
#### Step 1:
- Number of shaded triangles:
- In Step 1, the initial triangle is divided into 4 smaller congruent equilateral triangles, and the middle one is removed.
- Thus, the number of shaded triangles = 3.
- Area of each shaded triangle:
- Each of the 3 remaining triangles has 1/4 of the original area.
- Area = 256 / 4 = 64 square inches
#### Step 2:
- Number of shaded triangles:
- In Step 2, each of the 3 smaller triangles from Step 1 is divided again into 4 smaller equilateral triangles, removing the middle one from each group.
- So, the number of shaded triangles is 3 (previous step) 3 = 9.
- Area of each shaded triangle:
- Each of the new triangles has 1/4 of the area of the triangles from Step 1.
- Area = 64 / 4 = 16 square inches
#### Step 3:
- Number of shaded triangles:
- In Step 3, each of the 9 triangles from Step 2 is again divided and the middle triangle removed.
- Number of shaded triangles = 9 (previous step) 3 = 27.
- Area of each shaded triangle:
- Each of the new triangles now has 1/4 of the area of the triangles from Step 2.
- Area = 16 / 4 = 4 square inches
#### Step 4:
- Number of shaded triangles:
- In Step 4, the same process is repeated for each of the 27 triangles from Step 3.
- Number of shaded triangles = 27 (previous step) 3 = 81.
- Area of each shaded triangle:
- Each of the new triangles has 1/4 of the area of the triangles from Step 3.
- Area = 4 / 4 = 1 square inch
#### Step 5:
- Number of shaded triangles:
- In Step 5, repeat the process once more for each of the 81 triangles from Step 4.
- Number of shaded triangles = 81 (previous step) 3 = 243.
- Area of each shaded triangle:
- Each of the new triangles has 1/4 of the area of the triangles from Step 4.
- Area = 1 / 4 = 0.25 square inches
### Completed Table:
[tex]\[ \begin{array}{|l|l|l|} \hline \text{Step \#} & \begin{array}{l} \# \\ \text{of} \\ \text{shaded} \\ \text{triangle} \end{array} & \begin{array}{l} \text{Area of} \\ \text{each} \\ \text{shaded} \\ \text{triangle} \\ (\text{square inches}) \end{array} \\ \hline 0 & 1 & 256 \\ \hline 1 & 3 & 64.0 \\ \hline 2 & 9 & 16.0 \\ \hline 3 & 27 & 4.0 \\ \hline 4 & 81 & 1.0 \\ \hline 5 & 243 & 0.25 \\ \hline \end{array} \][/tex]
### Graphs:
1. Number of Shaded Triangles as a Function of the Step Number:
[tex]\[ \begin{array}{|c|c|} \hline \text{Step \#} & \# \text{ of Shaded Triangles} \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 27 \\ \hline 4 & 81 \\ \hline 5 & 243 \\ \hline \end{array} \][/tex]
Plot these points on a graph with the step number on the x-axis and the number of shaded triangles on the y-axis. The pattern shows exponential growth, specifically [tex]\(3^n\)[/tex] where [tex]\(n\)[/tex] is the step number.
2. Area of Each Shaded Triangle as a Function of the Step Number:
[tex]\[ \begin{array}{|c|c|} \hline \text{Step \#} & \text{Area (square inches)} \\ \hline 0 & 256 \\ \hline 1 & 64.0 \\ \hline 2 & 16.0 \\ \hline 3 & 4.0 \\ \hline 4 & 1.0 \\ \hline 5 & 0.25 \\ \hline \end{array} \][/tex]
Plot these points on a graph with the step number on the x-axis and the area of each shaded triangle on the y-axis. This pattern shows exponential decay, specifically [tex]\(\frac{256}{4^n}\)[/tex] where [tex]\(n\)[/tex] is the step number.
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