Answer :
a) Area of the rectangular garden A can be represented by the equation
A = f ( x ) = x ( 23 - x )
where x is the length of the rectangular fencing in feet
What is the Area of a Rectangle?
The area of the rectangle is given by the product of the length of the rectangle and the width of the rectangle
Area of Rectangle = Length x Width
Given data ,
Let the total length of the linear feet on fencing be = 46 feet
Now , Let the length of the rectangular fencing be = L
The width of the rectangular fencing be = W
So , the area of the rectangular fencing = Length x Width
Area of Rectangular fencing = L x W
And ,
when the length of the garden = 13 feet
Area of garden = 130 feet²
So , width of garden = Area of garden / length of garden
= 130 / 13
= 10 feet
And ,
when the length of the garden = 11.5 feet
Area of garden = 132.25 feet²
So , width of garden = Area of garden / length of garden
= 132.25 / 11.5
= 11.5 feet
So , the total length of the rectangular fencing = 46 feet
Perimeter of fencing = 46 feet
Perimeter P = 2 ( length + width )
= 2 x 23
= 46 feet
So , the equation to find the area of garden A = f ( x ) = x ( P/2 - x )
where P is the perimeter of the rectangular garden
Hence , Area of the rectangular garden A can be represented by the equation A = f ( x ) = x ( 23 - x )
where x is the length of the rectangular fencing in feet
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By using the perimeter formula a garden with a length and width of 11.5 feet each (forming a square) yields the maximum area of 132.25 square feet.
The student is tasked with determining the dimensions of a rectangular garden given a fixed amount of fencing material. If Lee has 46 linear feet of fencing, he can maximize the area of his garden by ensuring that the length and width of the garden provide the largest enclosed area.
According to the given information, if he sets one side (the length) to 13 feet, he obtains an area of 130 square feet, and with a length of 11.5 feet, he achieves a maximum area of 132.25 square feet.
To determine the width, we can use the fencing formula P = 2l + 2w, where P is the perimeter (46 feet), l is the length, and w is the width. Substituting the length values and solving for width, we find that:
- With length = 13 feet: 46 = 2(13) + 2w, which simplifies to 20 = 2w, and hence w = 10 feet.
- With length = 11.5 feet: 46 = 2(11.5) + 2w, which simplifies to 23 = 2w, and hence w = 11.5 feet, representing a square with maximum area.
Therefore, in Lee's case, forming a square with side lengths of 46/4 = 11.5 feet would provide the largest area, confirming the 132.25 square feet calculation.