A rectangle has a perimeter of 12 m. Express the area of the rectangle as a function of the length of one of its sides.

Let the length of one side of the rectangle be \( x \), and the other side be \( y \). The perimeter of a rectangle is given by:

\[
P = 2x + 2y
\]

Since the perimeter is 12 m, we can write:

\[
2x + 2y = 12
\]

Simplifying, we get:

\[
x + y = 6
\]

Now, express \( y \) in terms of \( x \):

\[
y = 6 - x
\]

The area \( A \) of a rectangle is the product of its length and width, so:

\[
A(x) = x \cdot y = x \cdot (6 - x)
\]

Therefore, the area as a function of \( x \) is:

\[
A(x) = 6x - x^2
\]

This is the required expression for the area in terms of the length of one side \( x \).

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. What value of \( x \) maximizes the area of the rectangle?
2. How does the area change if the perimeter is doubled?
3. What is the area when \( x = 3 \)?
4. How would the formula change if the perimeter were 20 m instead of 12 m?
5. Can you find the dimensions of the rectangle when the area is maximized?

**Tip:** For problems involving the perimeter of a rectangle, try simplifying the equation early to express one side in terms of the other.

Learn how to express the area of a rectangle as a function of one of its sides when given a perimeter of 12 m. This solution involves using algebraic concepts and quadratic functions. Suitable for students in Grades 8-10.

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