The equations model would be:
x(x + 8) = 256
The area of the playground needs to be 256 square yards.
The playground will be rectangular.
The city will pay to have soft pavement made of recycled tires installed in the playground.
In the first plan, one side 8 yards longer than the other side.
Let the width of the park be = w
So, the other side will be =
So, the area can be found as :
This equation will help to give the dimensions, related to area.
We can also solve this using quadratic formula.
We get dimensions as : 12.49 yards and 20.49 yards.
Part 1 : The area in terms of side length =
Area of square in terms of perimeter P :
Part 2: A simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards :
Part 1 : For any given perimeter P, the rectangle that encloses the greatest area is a square. . Write an equation for the area, A, in terms of the perimeter P, and the side length x.
We are given that the rectangle that encloses the greatest area is a square.
Let the side of the square be x
Let perimeter be P
So, perimeter of square'P' =
Area of square :
Thus the area in terms of side length =
Now to find area in terms of perimeter :
Now, Area of square :
So,Area of square in terms of perimeter P :
Part 2: Use the equation from Part I to result to write a simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards.
Since we have the equation of area in terms of perimeter :
Note: Perimeter tells the amount of fencing
Thus a simple equation to find the least amount of fencing necessary for a playground with an area of 256 square yards :
Length = 20.49 yards and Width = 12.49 yards.
The area of the rectangular playground is given by 256 yards square. It is also known that one of the sides of the playground is 8 yards longer than the other side. Therefore, let the smaller side by x yards. Then the longer side will be (x+8) yards. The area of the rectangle is given by:
Area of the rectangle = length * width.
256 = x*(x+8)
x^2 + 8x = 256. Applying the completing the square method gives:
(x)^2 + 2(x)(4) + (4)^2 = 256 + 16
(x+4)^2 = 272. Taking square root on both sides gives:
x+4 = 16.49 or x+4 = -16.49 (to the nearest 2 decimal places).
x = 12.49 or x = -20.49.
Since length cannot be negative, therefore x = 12.49 yards.
Since smaller side = x yards, thus smaller side = 12.49 yards.
Since larger side = (x+8) yards, thus larger side = 12.49+8 = 20.49 yards.
Thus, the length and the width to minimize the perimeter of fencing is 20.49 yards and 12.49 yards respectively!!!
Let the width of the rectangular park = x
As the area of the park is given to be 256 Sq yards
The perimeter of any rectangle is given as
P = 2(length * width )
In order to find the perimeter , we need to find the value of x first . In order to find x we have to use the relation which says that length is 8 yards more width of the park.
which will give us the quadratic equation. solving that we will get the value of x
Two sides would be
a₀ = 16 yards
a₁ = 16 yards
Since we have given that
Area of playground = 256 sq. yards
We need to find the dimensions of the rectangular area provides the least perimeter of fencing.
Since we know that square is a special kind of rectangle providing least perimeter with same area.
As we know the formula for Square :
Hence, two sides would be
a₀ = 16 yards
a₁ = 16 yards