A rectangular garden 50 m. long and 34 m. Wide is surrounded by a uniform dirt road. Find the width of the road if the total area of the garden and road is 4292 m^2.
garden: 1700 m^2
total: 4292 m^2
road: 2592 m^2
If the road has width w, its area is
2(50w + 34w) not counting the corners, which have area w^2 each. So,
2(50w+34w)+4w^2 = 2592
4w^2 + 168w - 2592 = 0
w^2 + 42w - 648 = 0
(w+54)(w-12) = 0
so, the road is 12 ft wide.
To find the width of the road, we need to know the area of both the garden and the road combined. The total area is given as 4292 m^2.
Let's denote the width of the road as 'x'. The length of the garden including the road would then be (50 + 2x) meters, and the width would be (34 + 2x) meters.
The area of the garden, excluding the road, is given by the product of the length and width: 50 * 34 = 1700 m^2.
To find the area of the road, we subtract the area of the garden from the total area: 4292 - 1700 = 2592 m^2.
Since the road surrounds the garden on all sides, its area is equal to the difference between the areas of the two rectangles:
(50 + 2x)(34 + 2x) - 1700 = 2592
Expanding the equation:
1700 + 100x + 68x + 4x^2 - 1700 = 2592
Combining like terms:
4x^2 + 168x - 2592 = 0
Dividing the equation by 4:
x^2 + 42x - 648 = 0
To solve this quadratic equation, we can use factoring or the quadratic formula. Factoring the equation, we get:
(x - 12)(x + 54) = 0
Therefore, x = 12 or x = -54.
The width of the road cannot be negative, so we take x = 12.
Hence, the width of the road is 12 meters.
To find the width of the road, we need to determine the area of the garden first.
The area of a rectangular garden can be calculated by multiplying its length by its width. In this case, the garden is 50 meters long and 34 meters wide, so the area of the garden is:
50 m * 34 m = 1700 m^2
The total area of the garden and road combined is given as 4292 m^2. This includes the area of the garden and the area of the road.
Let's assume the width of the road is 'x' meters. Since there is a road all around the garden, both the length and width of the garden will increase by 2x. Therefore, the dimensions of the garden including the road become:
Length = 50m + 2x
Width = 34m + 2x
Now, we can calculate the area of the garden and road combined:
Total Area = (Length + 2x) * (Width + 2x)
Substituting the values we know:
4292 m^2 = (50 m + 2x) * (34 m + 2x)
Expanding the equation:
4292 m^2 = (50 m * 34 m) + (50 m * 2x) + (34 m * 2x) + (2x * 2x)
Now, simplify the equation:
4292 m^2 = 1700 m^2 + 100 m * x + 68 m * x + 4 x^2
Rearranging the equation:
4 x^2 + 168 x + (1700 m^2 - 4292 m^2) = 0
Simplifying further:
4 x^2 + 168 x - 2592 = 0
Now we need to solve this quadratic equation to find the value of 'x'. We can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4, b = 168, and c = -2592:
x = (-168 ± √(168^2 - 4 * 4 * -2592)) / (2 * 4)
Calculating the discriminant (b^2 - 4ac):
Δ = 168^2 - 4 * 4 * -2592
Δ = 28224 + 41568
Δ = 69792
Now, substituting the values in quadratic formula:
x = (-168 ± √69792) / 8
Calculating the square root:
x ≈ (-168 ± 264.071) / 8
Simplifying further:
x ≈ (-168 + 264.071) / 8 or x ≈ (-168 - 264.071) / 8
x ≈ 96.071 / 8 or x ≈ -432.071 / 8
Simplifying:
x ≈ 12.009 or x ≈ -54.009
Since the width cannot be negative, the width of the road is approximately 12.009 meters.