Question

The area of the entire rectangle to the right is x(x+6) Find another expression for this area by finding the sum of the areas of the smaller rectangles.

Answer 1

To find another expression for the area of the rectangle \(x(x+6)\) by summing the areas of smaller rectangles, we can consider dividing the rectangle into smaller parts.

Let's say we break the rectangle into two smaller rectangles, where one rectangle has dimensions \(x\) (width) and 6 (height), and the other rectangle has dimensions \(x\) (width) and \(x\) (height).

1. **Area of the first smaller rectangle:**
- Width: \(x\)
- Height: \(6\)
- Area = \(x \cdot 6 = 6x\)

2. **Area of the second smaller rectangle:**
- Width: \(x\)
- Height: \(x\)
- Area = \(x \cdot x = x^2\)

Now, to find the total area of the rectangle by summing the areas of the smaller rectangles, we add the two areas together:

\[
\text{Total Area} = \text{Area of the first rectangle} + \text{Area of the second rectangle}
\]
\[
\text{Total Area} = 6x + x^2
\]

We can rearrange this expression as:

\[
x^2 + 6x
\]

Thus, another expression for the area of the rectangle \(x(x + 6)\) by finding the sum of the areas of the smaller rectangles is:

\[
x^2 + 6x
\]

This matches the original expression for the area \(x(x + 6)\), confirming that our partitioning and calculations are correct.