A recatangular yard is to be made with 120 feet fencing. The yard is divided into 4 equal parts. And an existing property fence will be used for one long side.
If x represents the width of the fence exeorsss its area A(x) in terms of x
Determine the dimensions of the rectangle that will make area maximum
width --- x
length --- y
If I understand your wording correctly, we will have
5x + 2y = 120 ----> y = 60 - 5x/2
A(x) = xy = x(60 - 5x/2) = 60x - (5/2)x^2
d A(x)/dx = 60 - 5x = 0 for a max of A(x)
5x = 60
x = 12
so the whole rectangle is 12 by 30 and has a max area of 360 ft^2
Ok so for the first part where it asks to express the area of width of the fence in terms of x is that part like this
A(x)=x(60-5x/2)
?
one long side is already there
5x + y = 120 ... y = 120 - 5x
area = x (120 - 5x^2) = 120x - 5x^2
the max is on the axis of symmetry ... x = -120 / (2 * -5) = 12
y = 60 ... area = 720
I'm confused now lol whose correct? 2 very different conclusions
The first answer has two long sides made of new fencing.
The second answer assumes that only one long side is new fencing.
By the way I saw a third way with three long sides and three short sides.
To solve this problem, we need to first setup the problem properly and then differentiate the area expression with respect to 'x' to find the maximum value.
Let's say the length of the yard is 'L' and the width is 'x'. Since the yard is divided into 4 equal parts, we have 4 sections of width 'x' each. So, the total length would be 4x.
We are given that the total fencing available is 120 feet. Since one long side is already fenced by an existing property fence, we can subtract the length of that fence from the total fencing available. This gives us:
2L + 5x = 120 - L
Simplifying this equation, we get:
3L + 5x = 120
Solving for L, we have:
L = (120 - 5x) / 3
Now, let's express the area A(x) in terms of 'x'. The area of a rectangle is given by the product of its length and width. So, the area A(x) can be expressed as:
A(x) = L * x
Substituting the value of L from the previously obtained equation, we get:
A(x) = ((120 - 5x) / 3) * x
Simplifying this expression, we have:
A(x) = (120x - 5x^2) / 3
To find the dimensions of the rectangle that will make the area maximum, we need to differentiate A(x) with respect to 'x' and set it equal to zero. Taking the derivative of A(x) and simplifying, we get:
A'(x) = (120 - 10x) / 3
Setting this derivative equal to zero and solving for 'x', we have:
(120 - 10x) / 3 = 0
120 - 10x = 0
10x = 120
x = 12
Therefore, the width of the fence that maximizes the area is 12.
To find the length, we can use the equation L = (120 - 5x) / 3 and substitute the value of x:
L = (120 - 5*12) / 3
L = 40
Therefore, the dimensions of the rectangle that will maximize the area are 12 feet (width) and 40 feet (length).