Question

a floor plan is given below.
which expression can be used to determine the area of the patio?
a ( 35r - 14r^3 )
b ( 4r^4 + 7r )
c ( 12r^5 + 24r^4 )
d ( 6r^5 + 6 )

Explanation:

Step1: Identify Patio's Dimensions

The patio is a rectangle. Its height is \( (3r + 6) \) and its width is \( 4r^4 \)? Wait, no, looking at the diagram, the patio's horizontal side (width) is \( 4r^4 \)? Wait, no, maybe I misread. Wait, the vertical side of the patio is \( (3r + 6) \), and the horizontal side (width) – looking at the family room and bedrooms, but the patio's width: wait, the patio is adjacent to the family room and bedrooms. Wait, no, the patio's dimensions: the left side is \( (3r + 6) \) (vertical), and the top side is \( 4r^4 \)? No, that can't be. Wait, maybe the patio is a rectangle with length \( (3r + 6) \) and width \( 4r^4 \)? No, wait, maybe the horizontal dimension (width) of the patio is the same as the sum of the family room's width and the bedrooms' width? No, the patio is on the left. Wait, the diagram: the patio is a red rectangle. The top of the patio is labeled \( 4r^4 \), and the left side is \( (3r + 6) \). Wait, no, maybe the patio's length is \( (3r + 6) \) (vertical) and width is \( 4r^4 \) (horizontal)? No, that seems odd. Wait, no, maybe the patio is a rectangle with length \( 4r^4 \) and height \( (3r + 6) \)? Wait, no, the area of a rectangle is length times width. Wait, maybe the patio's dimensions are: horizontal side (width) is \( 4r^4 \)? No, that doesn't make sense. Wait, looking at the family room: the family room's height is \( 7r \), and the bedrooms below have height \( 2r^2 \). Wait, maybe the patio's height is \( (3r + 6) \), and its width is \( 4r^4 \)? No, that can't be. Wait, maybe I made a mistake. Wait, the problem is to find the area of the patio. The patio is a rectangle. So area is length × width. Looking at the diagram, the patio's vertical side (height) is \( (3r + 6) \), and its horizontal side (width) is \( 4r^4 \)? No, that would be \( 4r^4 \times (3r + 6) \), but that's not one of the options. Wait, the options are: A. \( 35r - 14r^3 \), B. \( 4r^4 + 7r \), C. \( 12r^5 + 24r^4 \), D. \( 6r^3 + 6 \). Wait, let's check the options. Option C is \( 12r^5 + 24r^4 \), which is \( 4r^4 \times (3r + 6) \), because \( 4r^4 \times 3r = 12r^5 \) and \( 4r^4 \times 6 = 24r^4 \), so \( 4r^4(3r + 6) = 12r^5 + 24r^4 \), which is option C. Wait, maybe the patio's dimensions are length \( (3r + 6) \) and width \( 4r^4 \), so area is \( 4r^4 \times (3r + 6) \). Let's expand that: \( 4r^4 \times 3r + 4r^4 \times 6 = 12r^5 + 24r^4 \), which is option C.

Step2: Expand the Product

To find the area of the patio (a rectangle), we multiply its length and width. The length is \( 4r^4 \) and the width is \( (3r + 6) \). Using the distributive property (FOIL for monomial and binomial):
\( 4r^4 \times (3r + 6) = 4r^4 \times 3r + 4r^4 \times 6 \)
Simplify each term:
\( 4r^4 \times 3r = 12r^{4+1} = 12r^5 \)
\( 4r^4 \times 6 = 24r^4 \)
Combine the terms: \( 12r^5 + 24r^4 \)

Answer:

C. \( 12r^5 + 24r^4 \)