Question

Use a quadratic function to model the area of a rectangle if the sides are 2x+8 and x+1.

Answer 1

To model the area of a rectangle using a quadratic function, we can use the formula for the area of a rectangle, which is given by:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

In this case, the length is \( 2x + 8 \) and the width is \( x + 1 \). Thus, we can express the area \( A \) as follows:

\[
A = (2x + 8)(x + 1)
\]

Now, we will expand this expression to obtain a quadratic function.

1. First, distribute \( (2x + 8) \) on \( (x + 1) \):

\[
A = 2x(x) + 2x(1) + 8(x) + 8(1)
\]

2. Now perform the multiplications:

\[
A = 2x^2 + 2x + 8x + 8
\]

3. Combine the like terms:

\[
A = 2x^2 + (2x + 8x) + 8
\]
\[
A = 2x^2 + 10x + 8
\]

Thus, the area of the rectangle modeled as a quadratic function is:

\[
A(x) = 2x^2 + 10x + 8
\]