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?
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.
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
To maximize the total area of the enclosures, we need to find the dimensions (length and width) that will result in the largest area.
Let's assume the width of each enclosure is "w" meters.
Since the enclosures are identical, each enclosure will have two short sides, two long sides, and three sides along the river (which do not need to be fenced).
Given that the farmer has 1200 meters of fencing material, and each enclosure has four sides that need to be fenced, we have:
4w + 2 × 3 = 1200
4w + 6 = 1200
4w = 1200 - 6
4w = 1194
w = 1194 ÷ 4
w = 298.5
So, the width of each enclosure should be 298.5 meters.
To find the length of each enclosure, we know that each enclosure has two long sides and two short sides of equal length.
Let's assume the length of each enclosure is "l" meters.
The total length of the long sides will be 2 × l meters, and the total length of the short sides will be 2 × (l/2) meters.
Therefore, the total length of the fenced sides is:
2 × l + 2 × (l/2) = 4l
Since each enclosure has the same width and length, we have:
4l + w + w = 1200
4l + 298.5 + 298.5 = 1200
4l + 597 = 1200
4l = 1200 - 597
4l = 603
l = 603 ÷ 4
l = 150.75
So, the length of each enclosure should be approximately 150.75 meters.
To summarize, the dimensional (length and width) of each enclosed should be approximately 150.75 meters and 298.5 meters, respectively, in order to maximize the total area of the enclosures.
To determine the dimensions of each enclosure that will maximize the total area, we can follow these steps:
1. Visualize the problem: Draw a diagram representing the problem. Since the enclosures are rectangular and located along a straight river, the diagram should show three rectangles placed side by side, with the river forming the shared side.
2. Understand the constraints: We are given that the total fencing material available is 1200 meters, and the sides along the river do not need to be fenced. This means that the fencing material will only be used to enclose the three outer sides of the three identical rectangles.
3. Determine the variables and the objective function: Let's denote the width of each rectangle as "w" and the length as "l". The objective is to maximize the total area of the three rectangles, which is given by: Total Area = 3*(l - 2w)*w.
4. Express the objective function in terms of a single variable: To make it easier to maximize the area, we can express the objective function in terms of a single variable, either "l" or "w". Let's express it in terms of "w". The total area becomes: Total Area = 3*(1200 - 8w)*w.
5. Maximize the total area: To find the maximum value of the area, we need to take the derivative of the area function with respect to "w" and set it equal to zero. Then solve for "w".
d(Total Area) / d(w) = 3*(1200 - 8w) - 3*(l - 2w) = 0
Simplifying the equation, we get: 1200 - 8w - l + 2w = 0
Rearranging, we have: 1200 - 6w - l = 0
6. Substitute one variable with the other: Recall that the objective is to find the dimensions of the rectangle that maximize the area. Since we have two variables, "w" and "l", we need to express one in terms of the other. We can express "l" in terms of "w" using the above equation: l = 1200 - 6w.
7. Substitute the expression for "l" back into the total area equation: The total area becomes: Total Area = 3*(1200 - 8w)*w = 3*(1200 - 8w)*w = 3600w - 24w^2.
8. Find the maximum area: Differentiate Total Area with respect to "w" to find the critical points: d(Total Area) / d(w) = 3600 - 48w. Set this derivative equal to zero, and solve for "w":
3600 - 48w = 0
48w = 3600
w = 75
9. Calculate the length: Substitute the value of "w" back into the expression for "l" (l = 1200 - 6w):
l = 1200 - 6(75)
l = 1200 - 450
l = 750
10. Finalize the answer: Therefore, to maximize the total area, each enclosure should have a width of 75 meters and a length of 750 meters.
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
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.