1. Area of the original patio: 35 ft^2
The original patio has is a rectangle with length = 7 feet and width = 5 feet. The area of a rectangle is given by the product between length and width:
[tex]A=L \cdot W[/tex]
Therefore, since in this case L=7 and W=5, the area of the original patio is
[tex]A=(7 ft)(5 ft)=35 ft^2[/tex]
2. Area of section A: 7x ft^2
Section A is also a rectangle, with length = 7 feet and width = x. Therefore, the area of this section is equal to:
[tex]A=L\cdot W=(7 feet)(x)=7x[/tex]
3. Area of section B: 5x ft^2
Section B is also a rectangle, with length = x and width = 5 feet. Therefore, the area of this section is equal to:
[tex]A=L\cdot W=(x)(5 feet)=5x[/tex]
4. Area of section C: [tex]x^2 ft^2[/tex]
Section C is a square, with side equal to x. The area of a square is equal to the square of the length of the side:
[tex]A=L^2[/tex]
therefore, in this case, since L = x, the area of this section is
[tex]A=(x)^2 = x^2[/tex]
5. Total area of the new patio using addition: [tex]x^2 +12x+35[/tex] ft^2
The total area of the new patio is equal to the sum of the four areas calculated in the previous sections:
[tex]A=35 +7x +5x+x^2 = x^2 +12x+35[/tex] ft^2
6. Total area of the new patio using multiplication: [tex]x^2+12x+35[/tex]
The total area of the new patio is equal to the product between the length (7+x) and the width (5+x):
[tex]A=(7+x)(5+x)=35+7x+5x+x^2=x^2+12x+35[/tex] ft^2
7. Yes
As we can see by comparing the area calculated in 5. and the area calculated in 6., the two areas are equal.
8. 80 ft^2
We already have the formula for the area of the new patio:
[tex]A=x^2+12x+35[/tex]
If we substitute x=3, we find the value of the area:
[tex]A=(3)^2+12\cdot 3+35=9+36+35=80[/tex]
