A rectangle garden is to be surrounded by a walkway of constant width. The garden's dimensions are 30ft by 40ft. The total area, garden plus walkway, is to be 1800ft^2. What must be the width of the walkway to the nearest thousandth?
To determine the width of the walkway, we need to subtract the area of the garden from the total area of the garden plus walkway.
Given that the garden's dimensions are 30ft by 40ft, the area of the garden can be calculated by multiplying the length by the width:
Area of the garden = length × width = 30ft × 40ft = 1200ft^2
To find the width of the walkway, we subtract the area of the garden from the total area:
Width of walkway = Total area - Area of garden = 1800ft^2 - 1200ft^2 = 600ft^2
Since the walkway surrounds the garden on all sides, we can think of it as an enlarged rectangle. The width of the walkway will be the same on all sides, so let's call it x.
If we add the width of the walkway (x) to both the length and width of the garden, the dimensions of the enlarged rectangle including the walkway would be (30ft + 2x) by (40ft + 2x).
The area of the enlarged rectangle is given by multiplying the length by the width:
Area of enlarged rectangle = (30ft + 2x) × (40ft + 2x)
Now, we can set up an equation to solve for the width of the walkway:
(30ft + 2x) × (40ft + 2x) = Total area
Substituting in the known values:
(30ft + 2x) × (40ft + 2x) = 1800ft^2
Expanding the equation:
1200ft^2 + 60ftx + 80ftx + 4x^2 = 1800ft^2
Combining like terms:
4x^2 + 140ftx + 1200ft^2 - 1800ft^2 = 0
4x^2 + 140ftx - 600ft^2 = 0
Now, we have a quadratic equation that we can solve for x. We can either factor it or use the quadratic formula.
Factoring (if possible) or applying the quadratic formula will allow us to find the value of x, which represents the width of the walkway.