A farmer wants to make three identical rectangular enclosures (same width and same length ) along a straight river, if he has fencing material of 1200 meter and sides along the river need not fence, what should be the dimensional (length and width ) of each enclosed if the total area of the enclosure is to be maximum?

Answer 1

Let's assume that the length of each enclosure is x and the width is y. Since the farmer wants to make three identical enclosures, the total length of fencing used will be 4x (3 sides of each enclosure) and the total width of fencing used will be 2y (2 sides of each enclosure).

We know that the farmer has 1200 meters of fencing material, excluding the sides along the river. So,

4x + 2y = 1200 (equation 1)

We need to find the values of x and y that maximize the total area of three enclosures. The total area of three enclosures can be expressed as:

3xy

Using equation 1, we can express y in terms of x:

2y = 1200 - 4x
y = (600 - 2x)

Substituting this value of y in the area equation, we get:

A = 3x(600 - 2x)
A = 1800x - 6x^2

To maximize the area, we need to find the value of x that makes the derivative of A with respect to x equal to zero. So,

A' = 1800 - 12x
1800 - 12x = 0
x = 150

Substituting this value of x in equation 1, we get:

4(150) + 2y = 1200
y = 450

Therefore, the dimensions of each enclosure should be 150 meters by 450 meters, and the total area of three enclosures will be:

3(150)(450) = 303750 square meters

Answer 2

But your 4x+3y = 600+1350 = 1950
and he only has 1200 m of fence.

Also, it is not given whether the 3 pens are separate or share interior borders.

Answer 3

Apologies for the mistake in the previous answer. Here's a corrected solution:

Let's assume that the length of each enclosure is x and the width is y. Since the farmer wants to make three identical enclosures, the total length of fencing used will be 4x (3 sides of each enclosure) and the total width of fencing used will be y (1 side of each enclosure).

We know that the farmer has 1200 meters of fencing material, excluding the sides along the river. So,

4x + y = 1200 (equation 1)

We need to find the values of x and y that maximize the total area of three enclosures. The total area of three enclosures can be expressed as:

3xy

Using equation 1, we can express y in terms of x:

y = 1200 - 4x

Substituting this value of y in the area equation, we get:

A = 3x(1200 - 4x)
A = 3600x - 12x^2

To maximize the area, we need to find the value of x that makes the derivative of A with respect to x equal to zero. So,

A' = 3600 - 24x
3600 - 24x = 0
x = 150

Substituting this value of x in equation 1, we get:

4(150) + y = 1200
y = 600

Therefore, the dimensions of each enclosure should be 150 meters by 600 meters, and the total area of three enclosures will be:

3(150)(600) = 270000 square meters

Note that this solution assumes that the three pens share interior borders and there is no buffer space between them. If buffer space is required, the dimensions of each pen may need to be adjusted accordingly.

Answer 4

You are correct to get x=150, but you said y was the total length.
So the area would be just xy=90,000, not 3xy. If you want 3xy, then

If we let the length (x) be the sides parallel to the river, then
3x+4y = 1200
A = 3xy = 3x(300 - 3/4 x) = 900x - 9/4 x^2
then A is maximum at x=200
so each pen is 200 by 150, giving a total area of 90,000 m^2

Note that as always, maximum area is when the fencing is divided equally between lengths and widths, 600 m each:
3*200 = 600
4*150 = 600