A small garden measures 11 ft by 8 ft. A uniform border around the garden increases the total area to 154 ft2. What is the width of the border? I cant seem to get the right answer. Please help

(11+2w)(8+2w) = 154

You add 2w because both sides have a border.

Note that 154 = 14*11

To solve this problem, we can start by finding the area of the garden without the border.

The area of the garden can be calculated by multiplying the length by the width:
Area of the garden = Length of the garden * Width of the garden
= 11 ft * 8 ft
= 88 ft²

Let's assume the width of the border is 'w' feet.

If we add the border around the garden, the new dimensions of the garden including the border can be calculated as follows:

Length including the border = Length of the garden + 2 * Border width
= 11 ft + 2w ft

Width including the border = Width of the garden + 2 * Border width
= 8 ft + 2w ft

The area of the garden including the border can be calculated using these new dimensions:

Area of the garden including the border = (Length including the border) * (Width including the border)
= (11 ft + 2w ft) * (8 ft + 2w ft)
= 11 ft * 8 ft + 11 ft * 2w ft + 8 ft * 2w ft + 2w ft * 2w ft
= 88 ft² + 22w ft + 16w ft + 4w² ft²
= 88 ft² + 38w ft + 4w² ft²

We know that the area of the garden including the border is 154 ft², so we can set up an equation:

88 ft² + 38w ft + 4w² ft² = 154 ft²

Now let's solve this equation.

Rearranging terms and setting the equation equal to zero:

4w² ft² + 38w ft + 88 ft² - 154 ft² = 0

Combining like terms:

4w² ft² + 38w ft - 66 ft² = 0

Let's solve this quadratic equation using the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values for a, b, and c:

w = (-(38) ± √((38)^2 - 4(4)(-66))) / (2(4))
w = (-38 ± √(1444 + 1056)) / 8
w = (-38 ± √(2500)) / 8
w = (-38 ± 50) / 8

Now we have two possible solutions for w:

w1 = (-38 + 50) / 8
w1 = 12 / 8
w1 = 1.5 ft

w2 = (-38 - 50) / 8
w2 = -88 / 8
w2 = -11 ft

Since the width of the border cannot be negative, we can ignore the negative solution.

Therefore, the width of the border is 1.5 ft.

To find the width of the border, we need to first calculate the area of the garden without the border and then subtract it from the total area including the border. Here's how you can do it:

1. Calculate the area of the garden without the border:
- The length of the garden is 11 ft.
- The width of the garden is 8 ft.
- Multiply the length and width to find the area: 11 ft * 8 ft = 88 ft².

2. Subtract the area of the garden from the total area, which is 154 ft²:
- Total area with the border = 154 ft².
- Area of the garden without the border = 88 ft².
- Width of the border = Total area with the border - Area of the garden without the border.
So, Width of the border = 154 ft² - 88 ft² = 66 ft².

Therefore, the width of the border is 66 ft².