A gardener has 120 ft of fencing to fence in a rectangular garden. one side of the garden is bordered by a river and so it does not need any fencing.
1. what dimensions would guarantee a garden with an area of 1350 ft^2?
2. What dimensions would guarantee the greatest area? how much is the greatest area?
two width, each of x
one length of y
y + 2x = 120
y = 120 - 2x
area = xy
= x(120-2x)
= 120x - 2x^2
1. 120x - 2x^2 = 1350
x^2 - 60x + 675 = 0
(x - 45)(x + 15) = 0
x = 45 or x = -15, the last we will reject
if x = 45 , y = 30
2. area = 120 - 2x^2 is a parabola which opens downwards so the vertex will be the maximum .
the x of the vertex is -120/(-4) = 30
when x = 30
y = 120-60 = 60
A width of 30 and a length of 60 ft will produce the largest area of 1800 ft^2
To find the dimensions that would guarantee a garden with an area of 1350 ft², we can use the formula for the area of a rectangle:
Area = Length × Width
Since one side of the garden is bordered by a river and does not need fencing, let's call the length of the garden L and the width W.
Let's assume that the width W is the side facing the river, as it does not require fencing.
1. To guarantee a garden with an area of 1350 ft², we know that:
Area = 1350 ft²
Since the length L is not restricted by the available fencing, we can choose any value for it. Let's solve for the width W using the formula:
1350 ft² = L × W
We also know that the perimeter of the garden is 120 ft, and the two widths combined with the length gives us that perimeter.
Perimeter = 2L + W = 120 ft
We can rewrite this equation to solve for L in terms of W:
2L = 120 ft - W
L = (120 ft - W) / 2
Now we can substitute this value of L back into the area equation:
1350 ft² = L × W
1350 ft² = [(120 ft - W) / 2] × W
2700 ft² = (120 ft - W) × W
2700 ft² = 120W - W²
W² - 120W + 2700 = 0
Now we can use the quadratic formula to solve for W:
W = [-b ± √(b² - 4ac)] / (2a)
In this case, a = 1, b = -120, and c = 2700.
W = [-(-120) ± √((-120)² - 4(1)(2700))] / (2(1))
W = [120 ± √(14400 - 10800)] / 2
W = [120 ± √3600] / 2
W = [120 ± 60] / 2
W = (120 + 60) / 2 = 90 / 2 = 45
W = (120 - 60) / 2 = 60 / 2 = 30
So, the possible widths of the garden are 45 ft and 30 ft.
To find the corresponding lengths:
L = (120 ft - W) / 2
For W = 45 ft: L = (120 ft - 45 ft) / 2 = 75 ft / 2 = 37.5 ft
For W = 30 ft: L = (120 ft - 30 ft) / 2 = 90 ft / 2 = 45 ft
Therefore, the possible dimensions that would guarantee a garden with an area of 1350 ft² are 37.5 ft by 45 ft, or 45 ft by 30 ft.
Now let's move on to the second part of the question.
2. To find the dimensions that would guarantee the greatest area, we can use the same perimeter equation:
Perimeter = 2L + W = 120 ft
Solve this equation for L in terms of W:
2L = 120 ft - W
L = (120 ft - W) / 2
Now substitute this value of L into the area equation:
Area = L × W
Area = [(120 ft - W) / 2] × W
Area = (120W - W²) / 2
Area = 60W - 0.5W²
To find the maximum area, we need to find the highest point on the graph of this equation. The highest point occurs at the vertex of the parabola.
The vertex x-coordinate can be found using the formula: x = -b / (2a)
In this case, a = -0.5 and b = 60.
W = -60 / (2 × -0.5)
W = -60 / -1
W = 60
So, the width that would guarantee the greatest area is 60 ft.
To find the corresponding length:
L = (120 ft - W) / 2
L = (120 ft - 60 ft) / 2
L = 60 ft / 2
L = 30 ft
Therefore, the dimensions that would guarantee the greatest area are 60 ft by 30 ft. The greatest area is given by:
Area = Length × Width
Area = 30 ft × 60 ft
Area = 1800 ft²
To find the dimensions that guarantee a garden with an area of 1350 square feet, we can set up an equation using the formula for finding the area of a rectangle.
Let's assume the length of the garden is "L" and the width is "W". We are given that the garden has 120 feet of fencing, but since one side is bordered by a river and doesn't need fencing, we only need to fence three sides: two widths and one length.
The equation for the amount of fencing required is: 2W + L = 120.
Since we are looking for the dimensions that give an area of 1350 square feet, we can also set up an equation using the area formula: A = L * W = 1350.
Now, we can solve these two equations simultaneously to find the dimensions.
1. Solving for the guaranteed area of 1350 ft^2:
First, rearrange the fencing equation to solve for L: L = 120 - 2W.
Substitute this value of L into the area equation: (120 - 2W) * W = 1350.
Simplify: 120W - 2W^2 = 1350.
Rearrange the equation: 2W^2 - 120W + 1350 = 0.
We can solve this quadratic equation by factoring or by using the quadratic formula. After solving, we find that the width W is approximately 9.9 feet and the length L is approximately 100.2 feet. So, the dimensions that guarantee a garden with an area of 1350 sq ft are approximately 9.9 ft (width) by 100.2 ft (length).
2. To find the dimensions that guarantee the greatest area:
Since we know that the total amount of fencing is 120 ft, we can rewrite the fencing equation as: 2W + L = 120.
Now, let's isolate L in terms of W: L = 120 - 2W.
Substitute this value of L into the area equation: A = L * W = (120 - 2W) * W.
Expand the equation: A = 120W - 2W^2.
To find the dimensions that guarantee the greatest area, we need to find the maximum point of this quadratic equation. This occurs at the vertex, which has an x-coordinate of -b/2a (where a, b, and c are the coefficients of the quadratic equation).
In this case, a = -2, b = 120, and c = 0. Plugging these values into the formula, we get W = -120/(2 * -2) = 30.
Substitute this value of W back into the fencing equation to get L = 120 - 2(30) = 60.
Therefore, the dimensions that guarantee the greatest area are 30 ft (width) by 60 ft (length), and the greatest area is A = 30 * 60 = 1800 sq ft.