Question

You want to expand a small patio that is 5 feet by 7 feet as seen in the above diagram. You want to expand it by the same amount, x feet, on each side as shown in the diagram below.

What is the area of the original patio?

What is the area of section A?

What is the area of section B?

What is the area of section C?

The total area of the new patio is the sum of the areas. What is that sum?

Multiply the new area using the length (7+x) and width (5+x).

Are the two areas from 5. and 6. the same?

What is the area of the new patio if you expand it x = 3 feet?

Answer 1

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:

`A=L \cdot W`

Therefore, since in this case L=7 and W=5, the area of the original patio is

`A=(7 ft)(5 ft)=35 ft^2`

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:

`A=L\cdot W=(7 feet)(x)=7x`

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:

`A=L\cdot W=(x)(5 feet)=5x`

4. Area of section C:
`x^2 ft^2`

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:

`A=L^2`

therefore, in this case, since L = x, the area of this section is

`A=(x)^2 = x^2`

5. Total area of the new patio using addition:
`x^2 +12x+35` ft^2

The total area of the new patio is equal to the sum of the four areas calculated in the previous sections:

`A=35 +7x +5x+x^2 = x^2 +12x+35` ft^2

6. Total area of the new patio using multiplication:
`x^2+12x+35`

The total area of the new patio is equal to the product between the length (7+x) and the width (5+x):

`A=(7+x)(5+x)=35+7x+5x+x^2=x^2+12x+35` 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:

`A=x^2+12x+35`

If we substitute x=3, we find the value of the area:

`A=(3)^2+12\cdot 3+35=9+36+35=80`