Question

Use a quadratic function to model the area of a rectangle if the sides are 2x+2 and x+3.

Answer 1

To model the area of a rectangle as a quadratic function, we use the formula for the area \( A \) of a rectangle, which is given by:

\[
A = \text{length} \times \text{width}
\]

In this case, the sides of the rectangle are given as \( 2x + 2 \) (length) and \( x + 3 \) (width). Therefore, the area can be expressed as:

\[
A = (2x + 2)(x + 3)
\]

Now, we can expand this expression to obtain a quadratic function:

1. **Distribute** the terms:
\[
A = 2x(x) + 2x(3) + 2(x) + 2(3)
\]
\[
A = 2x^2 + 6x + 2x + 6
\]

2. **Combine like terms**:
\[
A = 2x^2 + 8x + 6
\]

Thus, the area \( A \) of the rectangle as a quadratic function is:

\[
A(x) = 2x^2 + 8x + 6
\]

This quadratic function models the area of the rectangle based on the variable \( x \).