A farmer plans to enclose a rectangular pasture adjacent to a river. (see figure). The pasture must contain 320,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river?
x = m
y = m
Plz help...thnks
To minimize the amount of fencing needed, we can choose the dimensions of the rectangular pasture to be as close to a square as possible.
Let's assume the length of the rectangle is x meters and the width is y meters.
Since the area of the rectangle needs to be 320,000 square meters, we have the equation:
x * y = 320,000
To minimize the amount of fencing, we want to minimize the perimeter of the rectangle. The perimeter is given by:
P = 2x + 2y
We can rewrite the equation for the area in terms of one variable using the fact that x * y = 320,000:
y = 320,000 / x
Substituting this expression for y in the perimeter equation, we get:
P = 2x + 2(320,000 / x)
To find the minimum perimeter, we can take the derivative of P with respect to x and set it equal to 0:
dP/dx = 2 - 2(320,000 / x^2) = 0
Simplifying the equation, we get:
2x^2 - 640,000 = 0
Dividing both sides by 2, we have:
x^2 - 320,000 = 0
Using the quadratic formula, the solutions for x are:
x = ±sqrt(320,000)
Since we are looking for positive dimensions, we take only the positive square root:
x = sqrt(320,000)
Calculating the value, we find:
x ≈ 565.68 meters
Now, we can substitute this value of x back into the equation for y:
y = 320,000 / x ≈ 320,000 / 565.68 ≈ 565.68 meters
Therefore, the dimensions that require the least amount of fencing are approximately:
x ≈ 565.68 meters
y ≈ 565.68 meters
To find the dimensions of the pasture that will require the least amount of fencing, we can use the concept of calculus and optimization.
Let's start by setting up the problem. We need to minimize the amount of fencing required, which means we need to minimize the length of the fencing while still enclosing the desired area of 320,000 square meters.
Let's assume the length of the pasture is x meters and the width (parallel to the river) is y meters. Since there is no fencing needed along the river, the width y will be equal to the length of the river.
Now, we can express the perimeter of the rectangular pasture in terms of x and y:
Perimeter = 2x + y
To eliminate y from the equation, we can rearrange the equation for the area of the rectangle:
Area = x * y = 320,000
Solving this equation for y, we get:
y = 320,000 / x
Now, we can substitute this expression for y into the perimeter equation:
Perimeter = 2x + 320,000 / x
To find the minimum perimeter, we need to find the critical points of this function. We can do this by taking the derivative of the perimeter equation with respect to x and setting it equal to zero:
d(Perimeter) / dx = 2 - 320,000 / x^2 = 0
Solving this equation, we find:
2x^2 - 320,000 = 0
x^2 = 320,000 / 2
x^2 = 160,000
x = sqrt(160,000)
x = 400 meters
Now, we can substitute this value of x into the equation for y to find the corresponding width:
y = 320,000 / x
y = 320,000 / 400
y = 800 meters
Therefore, the dimensions that will require the least amount of fencing are:
x = 400 meters
y = 800 meters
So, the pasture should be 400 meters long and 800 meters wide to minimize the amount of fencing required.
Since length x width = area
we have length=area/width
Let
w=width of the pasture, and
320000/w=length (parallel to river) of pasture
Total length of fencing
F=twice width + length
=2w + 320000/w
Find derivative of F with respect to w and equate to zero to find w.
i.e.
Solve for w in
dF/dw=0