Maria’s garden has 24 meters of fence and she wants to fence the rectangular garden with wall on one side of the garden. She needs no fence along the wall side. Find the largest area of the Maria’s farm that can be fenced.
let the length parallel to the wall by y m
let the other two lengths be x m
2x + y = 24
y = 24-2x
area = xy
= x(24-2x)
= -2x^2 + 24x
this is a downwards opening parabola, we have to find the vertex.
I don't know which method you use but the easiest way is to find the x of the vertex with -b/(2a)
= -24/(4)
= 6
when x=6 , y = 24-12 = 12
The long side has to be 12 m and each of the shorter sides 8 m
the maximum area is 6(12) = 72 m^2
Note , the sum of the 3 sides is 24
To find the largest area that can be fenced, we need to determine the dimensions of the rectangular garden that would maximize the area.
Let's assume the length of the rectangular garden is x meters, and the width is y meters.
According to the problem, Maria wants to fence all sides of the garden except the one side where there is already a wall. The perimeter of the garden would then be equal to the total length of fence she has, which is 24 meters.
Perimeter = 2 * (Length + Width) + Length
24 = 2 * (x + y) + x
24 = 2x + 2y + x
24 = 3x + 2y
We need to express one variable in terms of the other to find the maximum area. Let's rearrange the equation to solve for y:
y = (24 - 3x) / 2
Now, we can express the area of the rectangular garden in terms of x and y:
Area = Length * Width
Area = x * y
Area = x * ((24 - 3x) / 2)
To find the largest possible area, we need to maximize this equation. We can do this by finding the maximum point of the quadratic function.
To do that, we can take the derivative of the area equation with respect to x, and set it equal to 0:
d(Area)/dx = (24 - 3x)/2 - (3/2)x
0 = (24 - 3x)/2 - (3/2)x
Now, let's solve for x:
(24 - 3x)/2 - (3/2)x = 0
24 - 3x - 3x = 0
24 - 6x = 0
6x = 24
x = 4
Now, we can substitute this value into the area equation to find the maximum area:
Area = x * ((24 - 3x) / 2)
Area = 4 * ((24 - 3(4)) / 2)
Area = 4 * ((24 - 12) / 2)
Area = 4 * (12 / 2)
Area = 4 * 6
Area = 24
Therefore, the largest area that Maria can fence is 24 square meters.
To find the largest area of Maria's garden that can be fenced, we need to determine the dimensions of the garden that will maximize the area. Let's denote the length of the garden as L and the width as W.
We know that the fence will be placed on three sides (two widths and one length), while the garden will have a wall on one side (remaining length). Therefore, the total length of fence required will be L + 2W.
Given that Maria has 24 meters of fence, we can set up an equation:
L + 2W = 24
Now, we need to express the area of the garden in terms of L and W. The area of a rectangle is given by A = L x W.
We want to find the maximum area, so let's solve for L in terms of W using the equation above:
L = 24 - 2W
Substitute this expression for L in the area equation:
A = (24 - 2W) x W
Expand and simplify:
A = 24W - 2W^2
To find the maximum area, we need to find the vertex of the quadratic equation. The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a = -2, and b = 24.
W = -24 / (2 * -2)
W = 12
Now substitute this value of W back into the equation for L:
L = 24 - 2W
L = 24 - 2 * 12
L = 0
We obtained W = 12 and L = 0, but since the length cannot be zero, we should disregard this solution. It implies that the largest area occurs at the boundary, where W = 12.
Therefore, the largest area of Maria's garden that can be fenced is A = 12 x 12 = 144 square meters.
24/3 = 8
8 * 8 = 64 square meters