Arectangular lawn measuring 8m by 12m is surrounded by apaved path of constant width-w metres.The area of the path is 23m^2. Form aquadratic equation that must be satisfied by width and hence find the width of the path?
with of the whole garden = 8 + 2w
length of the whole garden = 12 + 2w
area of the whole thing = (8+2w)(12+2w)
area of the path = whole thing - area of lawn = 23
(8+2w)(12+2w) - (8)(12) = 23
expand, simplify and solve for w
To find the width of the path, we need to set up a quadratic equation using the given information.
Let's assume the width of the path is 'w' meters.
The total area of the rectangular lawn including the path can be calculated by adding the area of the lawn and the area of the path. Since the width is the same all around, we can express the length and width of the rectangular lawn as:
Length = 8m + 2w
Width = 12m + 2w
The area of the lawn is given by the product of the length and width:
Area of the lawn = (8m + 2w)(12m + 2w)
The area of the path is given as 23m^2. Since the path surrounds the lawn, the area of the path can be calculated by subtracting the area of the lawn from the total area:
Area of the path = Total area - Area of the lawn
We can now set up the equation:
23 = (8m + 2w)(12m + 2w) - (8m)(12m)
Now, let's simplify the equation.
23 = (8m + 2w)(12m + 2w) - 96m^2
Expanding the brackets gives us:
23 = 96m^2 + 16mw + 24mw + 4w^2 - 96m^2
Simplifying further:
23 = 40mw + 4w^2
Now, we have a quadratic equation:
4w^2 + 40mw - 23 = 0
This is the quadratic equation that must be satisfied by the width of the path.
To find the width of the path, we can use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 4, b = 40m, and c = -23. Plugging in these values into the quadratic formula will give us the two possible solutions for the width of the path.
To find the width of the path, we need to set up a quadratic equation using the given information.
Let's assume the width of the path is "w" meters.
We know that the length of the outer rectangle is increased by 2w (width on each side), and the width of the outer rectangle is increased by 2w (width on each end).
So, the length of the outer rectangle becomes (8 + 2w) meters, and the width becomes (12 + 2w) meters.
The area of the inner rectangle is 8m x 12m = 96m^2, and the area of the outer rectangle (including the path) is (8 + 2w)(12 + 2w) m^2.
The area of the path is the difference between the area of the outer rectangle and the area of the inner rectangle:
(8 + 2w)(12 + 2w) - 96 = 23
Simplifying the equation:
96 + 16w + 12w + 4w^2 - 96 = 23
Combine like terms:
4w^2 + 28w - 23 = 0
So, the quadratic equation that must be satisfied by the width of the path is:
4w^2 + 28w - 23 = 0
We can solve this equation to find the value of "w".