Question

A rectangle has a height of \[3\] and a width of \[3x^2+4x\].

Express the area of the entire rectangle.
Expression should be expanded.
An area model for a rectangle with a height of three and a width of three x squared plus four x. The rectangle is broken into two rectangles to isolate each term in the width. The first rectangle has a height of three and a width of three x squared. The second rectangle has a height of three and a width of four x. \[3x^2\]\[4x\]\[3\]
An area model for a rectangle with a height of three and a width of three x squared plus four x. The rectangle is broken into two rectangles to isolate each term in the width. The first rectangle has a height of three and a width of three x squared. The second rectangle has a height of three and a width of four x.
\[\text{Area} = \]

Answer 1

The area of a rectangle with a height of 3 and a width of 3x^2 + 4x is 9x^2 + 12x. We use the distributive property to multiply the height by each term in the width and then sum the products to get the expanded area of the rectangle.

Step-by-step explanation:

To express the area of the rectangle, we need to multiply the height by the width. The height is given as 3, and the width is given as 3x^2 + 4x. By using the distributive property, we can find the area as follows:

Area = height × width = 3 × (3x^2 + 4x) = 3 × 3x^2 + 3 × 4x = 9x^2 + 12x

Thus, the expanded area of the rectangle is 9x^2 + 12x.

To visualize this using an area model:

• The first rectangle has dimensions of 3 (height) and 3x^2 (width), so its area is 3 × 3x^2 = 9x^2.

• The second rectangle has dimensions of 3 (height) and 4x (width), so its area is 3 × 4x = 12x.

The total area is the sum of these two areas, which confirms that the area of the rectangle is indeed 9x^2 + 12x.