https://r2.erweima.ai/mathbot/image/1721550786333wuwcwb8t.jpg

To solve for the breadth of each shaded rectangle, we need to analyze the given information step-by-step.

### Given Data:
1. **Dimensions:**
   - Figure A: \(21 \text{ cm} \times \text{width}\)
   - Figure B: \(12 \text{ cm} \times \text{width}\)
   - Both figures contain two identical shaded rectangles within them.

2. **Area Relations:**
   - The total area of Figure A is 742 cm² more than the total area of Figure B.
   - The unshaded area of Figure A is \(\frac{5}{3}\) times the unshaded area of Figure B.

### Step-by-Step Solution:

1. **Total Area Calculation:**
   - Let the width of both figures be \( w \).
   - Total area of Figure A: \( 21w \)
   - Total area of Figure B: \( 12w \)
   - According to the problem, \( 21w = 12w + 742 \)
   - Solving for \( w \): 
     \[
     21w - 12w = 742 \\
     9w = 742 \\
     w = \frac{742}{9} \\
     w = 82.\overline{4} \text{ cm}
     \]

2. **Shaded and Unshaded Areas:**
   - Let the area of one shaded rectangle be \( ab \), where \( a \) is the breadth and \( b \) is the length.
   - Total shaded area in Figure A: \( 2ab \)
   - Total shaded area in Figure B: \( 2ab \)
   - Unshaded area in Figure A: \( 21w - 2ab \)
   - Unshaded area in Figure B: \( 12w - 2ab \)

3. **Unshaded Area Relation:**
   - According to the problem: \( 21w - 2ab = \frac{5}{3} \times (12w - 2ab) \)
   - Substituting \( w = 82.\overline{4} \):
     \[
     21 \times 82.\overline{4} - 2ab = \frac{5}{3} \times (12 \times 82.\overline{4} - 2ab)
     \]
   - Simplifying the equation:
     \[
     1732 - 2ab = \frac{5}{3} \times (989 - 2ab)
     \]
   - Let \( x = 2ab \):
     \[
     1732 - x = \frac{5}{3} \times (989 - x) \\
     1732 - x = \frac{5}{3} \times 989 - \frac{5}{3} x \\
     1732 - x = 1648.33 - 1.67x
     \]
   - Rearranging the terms to solve for \( x \):
     \[
     1732 - 1648.33 = -0.67x \\
     83.67 = 0.67x \\
     x = \frac{83.67}{0.67} \\
     x = 125
     \]
   - Thus, \( 2ab = 125 \Rightarrow ab = 62.5 \).

4. **Finding \( a \) (breadth):**
   - Since the total area of Figure A is divided into two identical shaded rectangles, we use the width \( w = 82.\overline{4} \) to get the specific values.
   - Given the dimensions of the rectangles, we can set \( b = 12 \text{ cm} \) (assuming the length from Figure B's unshaded part for simplicity, or you could choose another length for detailed calculation).

\[
ab = 62.5 \Rightarrow a \times 12 = 62.5 \\
a = \frac{62.5}{12} \\
a \approx 5.21 \text{ cm}
\]

Therefore, the breadth of each shaded rectangle is approximately \( 5.21 \text{ cm} \).

If you have any questions or need more details, please let me know. Here are some more questions you might want to consider:

1. How can we verify the unshaded areas relation calculation?
2. What if the length \( b \) is different?
3. How do we visualize the total area calculations in both figures?
4. How can the solution be cross-checked for accuracy?
5. What are the implications if the unshaded areas were different?

### Tip:
When solving geometry problems involving areas, always double-check the unit consistency and the correct substitution of known values.

Explore how to calculate the breadth of shaded rectangles in geometric figures using area calculations and linear equations. This problem involves analyzing dimensions and area relations between two figures with detailed step-by-step solutions. Suitable for Grades 10-12 students studying geometry and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation