A farmer wants to make three identical rectangular enclosures along a straight river, as in the diagram shown below. If he has 1680 yards of fence, and if the sides along the river need no fence, what should be the dimensions of each enclosure if the total area is to be maximized?

Well, it seems like this farmer wants to have some fancy enclosures along the river. I bet he wants to have a farm party and invite all the fish to join in!

To maximize the area with the given amount of fence, we need to figure out the dimensions of each enclosure. Since the sides along the river don't need any fence, let's focus on the other sides.

Let's call the length of each enclosure L and the width W. Since we have three enclosures, we'll need three lengths and three widths. Remember, we want all three enclosures to be identical.

Now, if we look at the diagram, we can see that the total length of fence needed will be 3L (for the lengths of the enclosures) and 4W (for the widths including the two ends).

According to the problem, the farmer has 1680 yards of fence. So, we can write down the equation:

3L + 4W = 1680.

To maximize the area, we want to find the dimensions that make the perimeter (3L + 4W) as small as possible. Let's try to minimize the perimeter first.

Now, I must be honest with you - as a clown, I'm not a math wiz, but I do love giving you a good laugh. And when it comes to minimizing perimeters, I would probably suggest making the enclosures super duper tiny, like a mouse-sized fence or something. But I don't think that's what the farmer wants!

In actuality, to get the maximum area, we need to maximize the width, while keeping the length as small as possible. That way, we can fit the most area within the given perimeter.

So, let's try something. Let's set W to be small, like 1 yard. That will make the 4W term in our equation (3L + 4W = 1680) small too. Now, we'll solve for L.

3L + 4(1) = 1680
3L + 4 = 1680
3L = 1676
L ≈ 558.67.

So, we've got one solution where the width is 1 yard and the length is approximately 558.67 yards. Remember, this is just one possible solution, but the dimensions of all three enclosures should be the same.

Now, keep in mind that maximizing the area is just one way to look at it. Maybe the farmer has some other plans, like building a mini water park or creating a maze for his ducks. So, don't be afraid to think outside the box and have a little fun with it!

Hope that made you chuckle a bit! Good luck with your math problem and have a fantastic day!

To maximize the total area of the three enclosures, we can assume that the width of each enclosure is x yards.

Since there are three enclosures, the length of each enclosure is 1680/3 = 560 yards (the remaining fence length divided equally among the three enclosures).

The total area of each enclosure is given by the formula: Area = length * width. Therefore, the area of each enclosure is A = 560x.

To find the dimensions that maximize the area, we need to find the maximum value of A.

Since we know that the fence along the river requires no fencing, the length of each enclosure should be perpendicular to the river. This means that we should choose the side with width x as the side parallel to the river.

Therefore, the dimensions of each enclosure should be 560 yards long and x yards wide.

To maximize the total area of the three identical rectangular enclosures, we need to find the dimensions of each enclosure. Let's break down the problem into steps:

Step 1: Understand the problem
We have three rectangular enclosures to be built along a straight river, and there is a fixed length of fence available (1680 yards). The sides of each enclosure that face the river don't require fencing.

Step 2: Define variables
Let's define the variables:
L: Length of each rectangular enclosure
W: Width of each rectangular enclosure

Step 3: Formulate equations
We need to find the dimensions (L and W) that maximize the total area of the three enclosures.

To solve this problem, we need to consider that the total length of the fence used will equal the sum of the perimeters of the three rectangular enclosures.
Total length of the fence = 3L + 4W (Three sides need fencing for each enclosure: two lengths and two widths)

Given that the total length of the fence is 1680 yards, we can write the equation:
3L + 4W = 1680

Step 4: Write the objective function
The objective is to maximize the total area of the three enclosures.
Total area of the three enclosures = 3 * L * W

Step 5: Solve the problem
To solve this optimization problem, we need to find the maximum value of the objective function by substituting the value of W from the constraint equation into the objective function equation.

From the constraint equation:
3L + 4W = 1680
4W = 1680 - 3L
W = (1680 - 3L)/4

Substitute the value of W in the objective function:
Total area of the three enclosures = 3 * L * [(1680 - 3L)/4]

Now, we need to find the value of L that maximizes the area. To find this, we can take the derivative of the objective function with respect to L and set it to zero:

d/dL [3L * (1680 - 3L)/4] = 0

This will give us the value of L that maximizes the area. Once we have L, we can substitute it back into the constraint equation to find the corresponding value of W.

Step 6: Solve for the dimensions
By solving the equation, you will find the values of L and W that maximize the total area of the three enclosures.

If each pen is x by y, with the x side parallel to the river, then

6x+4y = 1680

a = 3xy = 3*x(1680-6x)/4 = (2520x-9x^2)/2

This is just a parabola with vertex (hence maximum area) at x = 140

So, each pen is 140 by 210

As usual, the fencing is divided equally between lengths and widths.