the length of a rectangular garden is 8 feet longer than the width. The garden is surrounded by a 4-foot sidewalk. the sidewalk has an area of 320 square feet. find the dimensions of the garden.
please help
the garden is w by w+8, so the sidewalk has area
2(4w+4(w+8))+4*4^2 = 320
w = 12
The garden is 12x20
To find the dimensions of the garden, we can set up an equation based on the given information.
Let's assume the width of the garden is 'x' feet.
According to the given information, the length of the garden is 8 feet longer than the width. So, the length of the garden would be 'x + 8' feet.
The dimensions of the garden are then:
Width = x feet
Length = x + 8 feet
Now, let's consider the area of the garden.
The area of a rectangle is calculated by multiplying the length by the width.
Area of the garden = Length × Width
Area of the garden = (x + 8) × x
In the problem, it is stated that the garden is surrounded by a 4-foot sidewalk. So, to find the total area, we need to add the area of the garden to the area of the sidewalk.
Total Area = Area of the garden + Area of the sidewalk
Total Area = (x + 8) × x + 320
We know that the total area is 320 square feet. So, we can write the equation as:
320 = (x + 8) × x + 320
To solve this equation, we will simplify it step-by-step.
320 = x^2 + 8x + 320
Subtracting 320 from both sides, we get:
0 = x^2 + 8x
Now, let's factor out x from the equation:
0 = x(x + 8)
Setting each factor equal to zero, we find two possible solutions:
x = 0 or x + 8 = 0
The value 'x = 0' is not a valid solution because we cannot have a rectangular garden with zero width.
So, we focus on the second equation:
x + 8 = 0
Subtracting 8 from both sides, we get:
x = -8
Since the width of a garden cannot be negative, the solution 'x = -8' is not valid either.
Therefore, there are no real solutions for the width, and we cannot determine the dimensions of the garden based on the given information.
To find the dimensions of the garden, we will start by setting up equations based on the given information.
Let's say the width of the garden is "w" feet.
According to the problem, the length of the garden is 8 feet longer than the width. Therefore, the length of the garden can be represented as "w + 8" feet.
Now, let's consider the area of the garden without the sidewalk. It can be calculated by multiplying the width and length:
Area of the garden = width × length = w × (w + 8)
However, the problem states that the garden is surrounded by a 4-foot sidewalk. So, to find the total dimensions of the garden including the sidewalk, we need to add 8 feet (the length of the garden) to both the width and the length.
Width of the garden with the sidewalk = w + 2 × 4 = w + 8
Length of the garden with the sidewalk = w + 8 + 2 × 4 = w + 16
Now, let's calculate the area of the garden with the sidewalk:
Area of the garden with the sidewalk = (width with sidewalk) × (length with sidewalk) = (w + 8) × (w + 16)
According to the problem, the area of the sidewalk is 320 square feet. So we can set up the following equation:
Area of the sidewalk = Area of the garden with the sidewalk - Area of the garden without the sidewalk
320 = (w + 8) × (w + 16) - w × (w + 8)
Now, we can simplify the equation and solve for "w".
320 = (w^2 + 24w + 128) - (w^2 + 8w)
Simplifying further:
320 = w^2 + 24w + 128 - w^2 - 8w
This simplifies to:
320 = 16w + 128
Subtracting 128 from both sides:
192 = 16w
Dividing both sides by 16:
12 = w
So, the width of the garden is 12 feet.
Now, we can find the length of the garden:
Length of the garden = w + 8 = 12 + 8 = 20 feet
Therefore, the dimensions of the garden are:
Width = 12 feet
Length = 20 feet
length= 20 feet
width= 12 feet