Question
Use a quadratic function to model the area of a rectangle if the sides are 3x+3 and x+3.
Answers
GPT-4o mini
To model the area of a rectangle using a quadratic function, we first need to multiply the lengths of the two sides of the rectangle. The sides are given as \(3x + 3\) and \(x + 3\).
The area \(A\) of the rectangle can be calculated as:
\[
A = \text{Length} \times \text{Width} = (3x + 3)(x + 3)
\]
Now, let's expand this expression:
1. Distributing \(3x\):
\[
3x \cdot (x + 3) = 3x^2 + 9x
\]
2. Distributing \(3\):
\[
3 \cdot (x + 3) = 3x + 9
\]
Next, we combine both results:
\[
A = (3x^2 + 9x) + (3x + 9) = 3x^2 + 12x + 9
\]
Thus, the quadratic function that models the area of the rectangle is:
\[
A(x) = 3x^2 + 12x + 9
\]
The area \(A\) of the rectangle can be calculated as:
\[
A = \text{Length} \times \text{Width} = (3x + 3)(x + 3)
\]
Now, let's expand this expression:
1. Distributing \(3x\):
\[
3x \cdot (x + 3) = 3x^2 + 9x
\]
2. Distributing \(3\):
\[
3 \cdot (x + 3) = 3x + 9
\]
Next, we combine both results:
\[
A = (3x^2 + 9x) + (3x + 9) = 3x^2 + 12x + 9
\]
Thus, the quadratic function that models the area of the rectangle is:
\[
A(x) = 3x^2 + 12x + 9
\]