Please help with this Algebra Problem?

Question

A brick patio is twice as long as it is wide. It is bordered on all sides by a garden 1.5m wide. Find the dimensions of the patio if the area of the garden is 54m squared. Please show work

Answer 1

please draw a rectangle inside a bigger rectangle. The big rectangle is the garden border. The small rectangle is the patio.

It should look like a framed picture - the garden is the frame, the patio is the picture.

The border thickness is 1.5 m. Lets call the shorter side of the patio x and the longer side y. The shorter side of the garden is now 1.5+x+1.5 and the longer side of the garden is 1.5+y+1.5

Can you see 4 squares of 1.5*1.5 (one on each corner of the garden)?

So the area of the garden is the sum of
4 little squares 1.5*1.5
2 rectangles 1.5*x
2 rectangles 1.5*y

this sum you know already is equal to 54 sqm

so:
4*1.5*1.5 + 2*1.5*x + 2*1.5*y = 54

9 + 3x + 3y =54

3x + 3y = 45

3(x+y) = 45

x+y = 15

You know the longer side of the patio y is twice longer then the shorter side x, that is y = 2x

x+2x=15
3x=15
x=5

y=2x=2*5=10

sides of the patio 5 and 10

Answer 2

Let l = length and w = width.

l = 2w
w = w

So, the length of the garden would be 2w+3 and the width would be w+3. This is because the garden extends from the patio 1.5m in both directions each way.

(2w+3)(w+3)=54

2w^2 + 6w + 3w + 9 = 54

subtract 54 from both sides.

2w^2 + 9w - 45 = 0

Factor the polynomial....

(2w +15 ) ( w - 3) = 0 Product property of zero. Set each quantity equal to zero.

The roots of the polynomial are w = -7.5, which cannot be a dimension (we throw this answer out since length cannot be negative). Therefore the width of the patio is 3 m and the length is 6 m.

Answer 3

Let the width of the patio be W m and its length be 2W m.
The width of the garden is W + 3 and its length is 2W + 3.
So the area of the garden is
(W + 3) (2W + 3) = 54
=> 2W^2 + 3W + 6W + 9 = 54
=> 2W^2 + 9W - 45 = 0
=> (W - 3) (2W + 15) = 0
=> W = 3 or -7.5, but obviously W >= 0.
So the width is 3m and the length is 6m.

Answer 4

x = width and 2x = length of patio
so area of patio = x*2x = 2x^2

Area of garden plus patio is (x+3)(2x+3) = 2x^2+12x+9
So area of garden = 2x^2 +12x+9 - 2x^2 = 54
12x+9 =54
12x= 45
x = 45/12 = 3 .75m = patio width
2(3.75) = 7.5m = patio length