your father wants to fence a rectangular area in your backyard for a garden. one side of the garden is along rhe edgeof the yard is already fenced, so you only need to build a new fence along the other 3 sides of rectangle. if you have 80 meter of fencing available, what dimensions should the garden have in order to enclose the largest possible area?
To find the dimensions that will enclose the largest possible area, we can use the fact that a square has the maximum area for a given perimeter. However, since one side of the garden is already fenced, we need to find the dimensions for a rectangle.
Let's denote the length of the rectangle as L and the width as W. We have the following information:
- One side of the garden is already fenced, so the total length of fence used is L.
- The other two sides of the garden need to be fenced, so the total length of fence used is 2W.
We also know that the total length of fence available is 80 meters. Therefore, we can write the equation:
L + 2W = 80
To maximize the area, we can express the area A in terms of L and W:
A = L * W
Since we already have an equation expressing L in terms of W, we can substitute L = 80 - 2W into the equation for the area:
A = (80 - 2W) * W
To find the maximum area, we need to find the value of W that maximizes this equation. We can do this by finding the critical points of the equation, which occur when the derivative is equal to zero.
First, let's expand the equation for the area:
A = 80W - 2W^2
Now, let's find the derivative of the area with respect to W:
dA/dW = 80 - 4W
Setting this derivative equal to zero and solving for W, we have:
80 - 4W = 0
4W = 80
W = 20
Now that we have the value of W, we can substitute it back into the equation for L:
L = 80 - 2W
L = 80 - 2(20)
L = 80 - 40
L = 40
Therefore, the dimensions of the garden should be 40 meters by 20 meters in order to enclose the largest possible area.
To determine the dimensions that will enclose the largest possible area, we need to consider the properties of a rectangle. Let's assume the length of the rectangle is l meters, and the width is w meters.
Given that one side of the garden is already fenced, we only need to build new fences along the other three sides. Thus, the total length of the new fences required will be l + 2w.
According to the problem, we have 80 meters of fencing available. So, we can represent this information in an equation:
l + 2w = 80
To find the dimensions that will maximize the enclosed area, we need to express the area in terms of one variable so that we can differentiate it.
The formula for the area of a rectangle is A = l * w. In this case, we want to express the area, A, in terms of a single variable. We can rewrite the equation l + 2w = 80 as l = 80 - 2w, and substitute it into the area formula:
A = (80 - 2w) * w
Now, we can simplify the equation:
A = 80w - 2w^2
To find the maximum area, we need to find the maximum value of A. This can be achieved using calculus by taking the derivative of A with respect to w and setting it equal to zero:
dA/dw = 80 - 4w = 0
Solving for w, we find w = 20.
Substituting w = 20 back into the formula l + 2w = 80, we find l = 40.
Therefore, the garden should have dimensions of 40 meters by 20 meters in order to enclose the largest possible area with 80 meters of fencing.
you have 2x+y=80
and you want the maximum area,
a = xy = x(80-2x)
find the vertex of that parabola for max area. You will find that the fence is equally distributed among the lengths and widths.
That is,
2x = 40
y = 40
so the max area pen is 20x40