Question

use the diagram to answer the questions.

diagram of a house floor plan with labeled rooms and algebraic expressions for lengths and widths

1. what is the area of the dining room?
2. what is the area of the kitchen?
3. what is the combined area of the bedrooms?
4. what is the area of the entire house?

Explanation:

Question 1: Area of Dining Room

Step1: Identify length and width

The length of the dining room is \( x + 12 \) and the width is \( 2x^2 - 4x + 1 \).

Step2: Multiply length and width

To find the area, we multiply the length and the width: \( (x + 12)(2x^2 - 4x + 1) \)
Expand the product:
\[

$$\begin{align*} &x(2x^2 - 4x + 1) + 12(2x^2 - 4x + 1)\\ =& 2x^3 - 4x^2 + x + 24x^2 - 48x + 12\\ =& 2x^3 + 20x^2 - 47x + 12 \end{align*}$$

\]

Step1: Identify length and width

The length of the kitchen is \( x^2 + 5 \) and the width is \( 3x + 2 \).

Step2: Multiply length and width

To find the area, we multiply the length and the width: \( (x^2 + 5)(3x + 2) \)
Expand the product:
\[

$$\begin{align*} &x^2(3x + 2) + 5(3x + 2)\\ =& 3x^3 + 2x^2 + 15x + 10 \end{align*}$$

\]

Step1: Area of Bedroom A

The length of Bedroom A is \( x^2 + 4 \) and the width is \( 2x^2 + 2x - 1 \).
Area of Bedroom A: \( (x^2 + 4)(2x^2 + 2x - 1) \)
Expand:
\[

$$\begin{align*} &x^2(2x^2 + 2x - 1) + 4(2x^2 + 2x - 1)\\ =& 2x^4 + 2x^3 - x^2 + 8x^2 + 8x - 4\\ =& 2x^4 + 2x^3 + 7x^2 + 8x - 4 \end{align*}$$

\]

Step2: Area of Bedroom B

The length of Bedroom B is \( 2x^2 - 10 \) and the width is \( 3x + 2 \).
Area of Bedroom B: \( (2x^2 - 10)(3x + 2) \)
Expand:
\[

$$\begin{align*} &2x^2(3x + 2) - 10(3x + 2)\\ =& 6x^3 + 4x^2 - 30x - 20 \end{align*}$$

\]

Step3: Combine the areas

Add the area of Bedroom A and Bedroom B:
\[

$$\begin{align*} &(2x^4 + 2x^3 + 7x^2 + 8x - 4) + (6x^3 + 4x^2 - 30x - 20)\\ =& 2x^4 + (2x^3 + 6x^3) + (7x^2 + 4x^2) + (8x - 30x) + (-4 - 20)\\ =& 2x^4 + 8x^3 + 11x^2 - 22x - 24 \end{align*}$$

\]

Answer:

\( 2x^3 + 20x^2 - 47x + 12 \)

Question 2: Area of Kitchen