Suppose a farmer has 1,000 feet of fence and wishes to build two identical rectangular enclosures. What should be the dimensions of each enclosure if the total area is to be a maximum?
Set problem up and solve using derivatives.
To find the dimensions that will maximize the total area of the two rectangular enclosures, we can set up the problem using optimization techniques and solve it using derivatives.
Let's assume the length of each enclosure is 'x' feet, and the width is 'y' feet. Since there are two identical enclosures, we can consider one of them to derive the equation for the total area.
The total length of fencing used will be the sum of the perimeters of the two enclosures: 2x + 2y.
Given that the farmer has 1,000 feet of fence, we can write the equation:
2x + 2y = 1000
Now, the total area of the two enclosures can be expressed as the product of their lengths and widths:
Total Area, A = 2xy
To maximize the area, we need to express the area in terms of a single variable. So, we can rewrite the equation as:
y = (1000 - 2x) / 2
Now, substitute this expression for 'y' in the area equation to get it in terms of 'x' only:
A = 2x * [(1000 - 2x) / 2]
Simplifying further, we have:
A = x * (1000 - 2x)
To find the dimensions that maximize the area, we need to find the value of 'x' that maximizes the function A.
To do that, we can take the derivative of A with respect to 'x' and set it to zero:
dA/dx = 1000 - 4x = 0
Solving this equation, we find:
4x = 1000
x = 1000 / 4
x = 250 feet
Substitute this value of 'x' back into the equation for 'y' to find its value:
y = (1000 - 2x) / 2
y = (1000 - 2 * 250) / 2
y = (1000 - 500) / 2
y = 500 / 2
y = 250 feet
Therefore, the dimensions of each rectangular enclosure that will maximize the total area are:
Length: 250 feet
Width: 250 feet
To solve this problem, we need to set up an equation representing the area of the rectangular enclosures and then find the maximum of that equation using derivatives.
Let's assume that the length of each enclosure is x feet. Since we have two identical enclosures, the total length of the fences used is 2x.
The remaining 1000 - 2x feet of the fence will be used for the two widths of the enclosures. Therefore, each width will be (1000 - 2x) / 2 = 500 - x feet.
Now, we can calculate the area of the rectangular enclosure. Area = length * width = x * (500 - x).
To find the maximum area, we need to take the derivative of the area equation with respect to x and set it equal to zero.
d/dx (x * (500 - x)) = 500 - 2x = 0.
Solving this equation, we find x = 250.
So, the length of each enclosure should be 250 feet, and the width should be 500 - 250 = 250 feet.
Therefore, each enclosure should have dimensions 250 feet by 250 feet to maximize the total area.
if the sides are x,y
a = 2xy
now, 4x+4y=1000, so y=250-x and so
a = 2x(250-x) = 500x - 2x^2
da/dx = 500-4x
da/dx=0 when x=125
so, the enclosures are both squares, 125x125