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The L-shaped bracket shown below has an overall length of L=6.2cm and an overall width of W = 3.9 cm. Find x if the area of the bracket is 15.8cm^2.

We are given an L-shaped bracket with:

- Total length \( L = 6.2 \, \text{cm} \)
- Total width \( W = 3.9 \, \text{cm} \)
- Area of the bracket \( A = 15.8 \, \text{cm}^2 \)
- The small cutout corners labeled as \( x \), which need to be determined.

### Step 1: Define the area of the bracket

The L-shaped bracket can be thought of as a large rectangle with two small squares removed, both of size \( x \times x \). 

1. The total area of the larger rectangle (before cutting out the small squares) is:
   \[
   A_{\text{total}} = L \times W = 6.2 \, \text{cm} \times 3.9 \, \text{cm} = 24.18 \, \text{cm}^2
   \]

2. The two small squares removed have a combined area of:
   \[
   A_{\text{removed}} = 2 \times x^2
   \]

Thus, the area of the bracket is the total area minus the area of the removed squares:
\[
A_{\text{bracket}} = A_{\text{total}} - A_{\text{removed}} = 24.18 - 2x^2
\]

### Step 2: Set up the equation for the area

Since we know that the area of the bracket is 15.8 cm², we can set up the following equation:
\[
24.18 - 2x^2 = 15.8
\]

### Step 3: Solve for \( x \)

Now, solving the equation:
\[
24.18 - 15.8 = 2x^2
\]
\[
8.38 = 2x^2
\]
\[
x^2 = \frac{8.38}{2} = 4.19
\]
\[
x = \sqrt{4.19} \approx 2.05
\]

So, \( x \approx 2.05 \, \text{cm} \).

### Conclusion

The value of \( x \) is approximately 2.05 cm.

Do you need any further details or have any questions?

### Related questions:
1. How would the result change if the area of the bracket was 20 cm²?
2. How can you check if the area equation was set up correctly?
3. What are other shapes where a similar area calculation applies?
4. How would you express the total perimeter of the L-shape given \( x \)?
5. Can you calculate the area if only one square cutout was removed?

### Tip:
When dealing with composite shapes like this one, break the problem into smaller, manageable parts (like total area minus cutouts). It makes the solution easier to handle step by step.

The L-shaped bracket shown has an overall length of L=6.2cm and an overall width of W = 3.9 cm. Find x if the area of the bracket is 15.8cm^2.

Learn how to calculate the value of x for an L-shaped bracket with an overall length of 6.2 cm, width of 3.9 cm, and a total area of 15.8 cm². This problem covers geometric concepts and algebraic formulas suitable for Grades 9-10.

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