To find the area of the rectangle, we need to use the given information: the perimeter is 52 cm, the length is $3x - 1$, and the width is $2x + 3$.

Step 1: Find the value of $x$

The formula for the perimeter of a rectangle is:

$\text{Perimeter} = 2 \times (\text{Length} + \text{Width})$

Substitute the given values:

$52 = 2 \times ((3x - 1) + (2x + 3))$

Simplify the expression inside the parentheses:

$52 = 2 \times (5x + 2)$

Divide both sides by 2:

$26 = 5x + 2$

Solve for $x$ by subtracting 2 from both sides:

$24 = 5x$

Then, divide by 5:

$x = \frac{24}{5} = 4.8$

Step 2: Find the Length and Width

Now, substitute $x = 4.8$ back into the expressions for the length and width:

$\text{Length} = 3x - 1 = 3(4.8) - 1 = 14.4 - 1 = 13.4 \, \text{cm}$

$\text{Width} = 2x + 3 = 2(4.8) + 3 = 9.6 + 3 = 12.6 \, \text{cm}$

Step 3: Find the Area

The area $A$ of a rectangle is given by:

$A = \text{Length} \times \text{Width}$

Substitute the values for length and width:

$A = 13.4 \times 12.6 = 168.84 \, \text{cm}^2$

Conclusion

The area of the rectangle is 168.84 cm².

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Would you like more details, or do you have any questions?

Related Questions:

1. What would be the area if the length and width were swapped?
2. How would the area change if the perimeter were 60 cm instead of 52 cm?
3. How do you derive the formula for the perimeter of a rectangle?
4. What is the relationship between the perimeter and area of a rectangle?
5. Can the value of $x$ be a different number for the same perimeter?

Tip:

Always double-check your values by substituting them back into the original equations to ensure they satisfy all given conditions.