Two rectangular picture frames are mathematically similar.
(a) The areas of the frames are 350 cm2 and 1134 cm2 . The width of the smaller frame is 17.5 cm.
Calculate the width of the larger frame.

To solve the problem, we need to use the property of similar figures. For two similar rectangles, the ratio of their areas is the square of the ratio of their corresponding lengths. Let's denote the width of the smaller frame as \( w_1 \) and the width of the larger frame as \( w_2 \). 

Given:
- \( A_1 = 350 \) cm² (area of the smaller frame)
- \( A_2 = 1134 \) cm² (area of the larger frame)
- \( w_1 = 17.5 \) cm (width of the smaller frame)

We need to find \( w_2 \), the width of the larger frame.

The ratio of the areas of the two frames is:
\[
\frac{A_2}{A_1} = \frac{1134}{350}
\]

Simplifying this ratio:
\[
\frac{1134}{350} = \frac{1134 \div 2}{350 \div 2} = \frac{567}{175}
\]

To find the ratio of the widths, we take the square root of the area ratio because the areas of similar rectangles are proportional to the square of the lengths:
\[
\frac{w_2}{w_1} = \sqrt{\frac{A_2}{A_1}} = \sqrt{\frac{567}{175}}
\]

Calculating the square root:
\[
\sqrt{\frac{567}{175}} = \sqrt{3.24} \approx 1.8
\]

Now, we can find \( w_2 \):
\[
w_2 = w_1 \times 1.8 = 17.5 \times 1.8 = 31.5 \, \text{cm}
\]

Thus, the width of the larger frame is:
\[
\boxed{31.5 \, \text{cm}}
\]

Would you like further details or have any questions about this solution?

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Here are some related questions you might find interesting:
1. What is the ratio of the perimeters of the two frames?
2. If the length of the smaller frame is known, how can we find the length of the larger frame?
3. How would the problem change if the given dimensions were in inches instead of centimeters?
4. Can the relationship between area and width be applied to circles or other shapes?
5. What is the significance of the aspect ratio in similar figures?
6. How can we find the diagonal of the larger frame if the diagonal of the smaller frame is known?
7. What is the effect on the area if both the width and length of a frame are doubled?
8. How can this principle of similarity be used in real-life applications, such as resizing images?

**Tip:** When dealing with similar figures, always remember that the ratio of the areas is the square of the ratio of the corresponding sides.

Learn how to calculate the width of a larger picture frame using the properties of similar rectangles. Given the areas of the frames and the width of the smaller frame, find the width of the larger frame using the ratio of areas and the square root of the ratio of their corresponding sides. Suitable for Grades 7-9.

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