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Let's break down the problem step by step:

### Part (a): 
We are given three sets of dimensions for rectangles, and we need to check if they are golden rectangles. The golden rectangle has a length-to-width ratio of approximately **1.62:1**. To determine if a rectangle is golden, we will calculate the ratio of length to width for each rectangle and compare it to **1.62**.

#### First rectangle: 24 m × 38.9 m
\[
\text{Ratio} = \frac{38.9}{24} \approx 1.62
\]
This matches the golden ratio, so **this is a golden rectangle**.

#### Second rectangle: 52 cm × 120.5 cm
\[
\text{Ratio} = \frac{120.5}{52} \approx 2.32
\]
This is not close to the golden ratio, so **this is not a golden rectangle**.

#### Third rectangle: 348 mm × 565 mm
\[
\text{Ratio} = \frac{565}{348} \approx 1.62
\]
This matches the golden ratio, so **this is a golden rectangle**.

### Part (b):
We are given the width of a golden rectangle as **6.4 m** and need to calculate its length using the golden ratio.

The golden ratio is \( 1.62 \), so:
\[
\text{Length} = 1.62 \times 6.4 = 10.368 \text{ m}
\]
Rounded to the nearest tenth, the length is **10.4 m**.

### Summary of answers:
- **(a)**: The golden rectangles are: **24 m × 38.9 m** and **348 mm × 565 mm**.
- **(b)**: The length of the rectangle is **10.4 m**.

Would you like further details on any step or any related explanations?

Here are 5 related questions to expand your understanding:
1. How is the golden ratio derived, and why is it important in art and architecture?
2. Can you find other famous structures that incorporate the golden rectangle in their design?
3. How would the calculation differ if the ratio was not 1.62 but another value, like 1.5?
4. How do you convert units (e.g., meters to centimeters) when working with dimensions of rectangles?
5. Why does rounding the length to the nearest tenth make sense in practical architectural design?

**Tip**: The golden ratio appears not just in geometry but also in nature, such as in the arrangement of leaves and the spirals of shells. Studying it can provide insights into both mathematics and aesthetics!

The golden rectangle is used often in art and architecture. Which of these dimensions of rectangles are examples of golden rectangles? If the width of a golden rectangle is 6.4 m, what is its length? Give your answer to the nearest tenth of a meter.

Learn how to determine whether given rectangle dimensions fit the golden ratio (1.62:1) and how to calculate the length of a golden rectangle when the width is known. Explore geometry concepts and practical examples of the golden rectangle in art and architecture.

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