This problem involves two similar rectangles, and we are asked to find the perimeter of the second rectangle based on the perimeter of the first rectangle.

Steps to solve:

1. Perimeter of rectangle formula: The perimeter $P$ of a rectangle is given by: $P = 2 \times (\text{length} + \text{width})$

2. Scaling factor: Since the rectangles are similar, the sides of the second rectangle are scaled by the same factor. The ratio of corresponding sides of the rectangles is constant. From the image, we know:

   • The length of the first rectangle is 10 yards.
   • The length of the second rectangle is 160 yards.

   The scaling factor $k$ can be found as the ratio of the lengths: $k = \frac{\text{length of second rectangle}}{\text{length of first rectangle}} = \frac{160}{10} = 16$

3. Using the perimeter of the first rectangle:

   • The perimeter of the first rectangle is 34 yards.
   • Let the width of the first rectangle be $w_1$. Then, the perimeter formula becomes: $34 = 2 \times (10 + w_1)$
   • Solving for $w_1$: $34 = 20 + 2w_1 \quad \Rightarrow \quad 2w_1 = 14 \quad \Rightarrow \quad w_1 = 7$ So, the width of the first rectangle is 7 yards.

4. Scaling the width for the second rectangle: The width of the second rectangle $w_2$ is scaled by the same factor $k = 16$: $w_2 = 16 \times w_1 = 16 \times 7 = 112 \text{ yards}$

5. Calculating the perimeter of the second rectangle: The perimeter of the second rectangle is: $P_2 = 2 \times (160 + 112) = 2 \times 272 = 544 \text{ yards}$

Thus, the perimeter of the second rectangle is 544 yards.

Would you like further details on any part of this process, or is there something specific you'd like me to explain? Here are some follow-up questions:

1. How do we calculate the perimeter of any rectangle?
2. What is the concept of similar figures in geometry?
3. Can the scaling factor be used for areas or volumes as well?
4. How do you use ratios to find missing dimensions in similar shapes?
5. What are some real-world applications of similar shapes?

Tip: Always ensure you clearly understand the relationship between corresponding parts of similar figures (like sides, perimeters, areas) before applying formulas!