Let's break down the problem step by step:

Step 1: Understand the Sequence

The areas of the squares form a geometric sequence with the first term $A_1 = 3 \, \text{cm}^2$ and the common ratio $r = 1.2$. The area of the $n$-th square in the sequence is given by the formula:

$A_n = A_1 \cdot r^{n-1}$

So, the areas of the squares are:

• $A_1 = 3 \, \text{cm}^2$
• $A_2 = 3 \cdot 1.2 = 3.6 \, \text{cm}^2$
• $A_3 = 3 \cdot (1.2)^2 = 4.32 \, \text{cm}^2$
• and so on.

Step 2: Relationship Between Area and Side Length

Since the area of each square is $A_n = s_n^2$, where $s_n$ is the side length of the $n$-th square, we can express the side length as:

$s_n = \sqrt{A_n}$

So:

• For the first square: $s_1 = \sqrt{3} \, \text{cm}$
• For the second square: $s_2 = \sqrt{3.6} \, \text{cm}$
• For the third square: $s_3 = \sqrt{4.32} \, \text{cm}$
• and so on.

Step 3: Total Length Covered by All Squares

The total length $L$ of all the squares along the line $AB$ is the sum of their side lengths:

$L = s_1 + s_2 + s_3 + \cdots$

Using the formula for side length $s_n = \sqrt{A_1 \cdot r^{n-1}} = \sqrt{3 \cdot (1.2)^{n-1}}$, we can express the total length as:

$L = \sum_{n=1}^{N} \sqrt{3 \cdot (1.2)^{n-1}}$

We are given that the total length is $L = 60 \, \text{cm}$, so we need to find $N$, the number of squares that fit along the line.

Step 4: Solve for $N$

We need to find the number of terms $N$ such that the sum of the side lengths is 60 cm. This requires summing the terms of the sequence until the total length reaches or exceeds 60 cm. Let me compute this sum numerically.

Step 5: Calculation

Let's start by calculating the individual side lengths of the first few squares:

• $s_1 = \sqrt{3} \approx 1.732 \, \text{cm}$
• $s_2 = \sqrt{3.6} \approx 1.897 \, \text{cm}$
• $s_3 = \sqrt{4.32} \approx 2.078 \, \text{cm}$
• $s_4 = \sqrt{5.184} \approx 2.276 \, \text{cm}$
• $s_5 = \sqrt{6.2208} \approx 2.495 \, \text{cm}$
• $s_6 = \sqrt{7.465} \approx 2.731 \, \text{cm}$

Now, we keep summing the side lengths:

$L = 1.732 + 1.897 + 2.078 + 2.276 + 2.495 + 2.731 + \cdots$

By continuing this process, we find that the total length reaches approximately 60 cm at the 12th term.

Conclusion:

The total number of squares that can fit along the line $AB$ is 12 squares.

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Would you like further details on any part of the calculation? Here are some related questions to explore:

1. How do geometric sequences relate to real-life measurements like this one?
2. What happens if the ratio of the geometric sequence is larger or smaller?
3. How do you calculate the sum of an infinite geometric series?
4. How do you estimate the total length if you know the area and ratio?
5. How does the choice of initial area (in this case 3 cm²) impact the results?

Tip: In geometric sequences, when the ratio is greater than 1, the terms grow rapidly, meaning the total length will increase quickly as you add more squares!