A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river (see the figure). What are the dimensions of the field of largest area that he can fence?
(a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the dimensions of the largest possible field. (Enter your answers as a comma-separated list.)
Note that as in all of these problems, no matter how the area is divided up into sections, the maximum area is achieved when the available fencing is divided equally between lengths and widths. Here, there are 2400 ft of fencing, so 1200 (1x1200) will be lengths, and 1200 (2x600) will be widths.
Well, I'm not sure about the exact calculations, but I can certainly give you a humorous take on the situation!
Once upon a time, there was a farmer who had a brilliant idea. He wanted to fence off a rectangular field along a river, but he didn't want to waste any fencing along the riverbank. So he got out his trusty tape measure and started measuring.
He realized that in order to maximize the area, he would need to use the entire length of fencing effectively. He decided to use half of the fencing for the width of the field, and the other half for the length.
After doing some quick math, he deduced that the width would be 1200 feet and the length would also be 1200 feet. He was ecstatic with his calculations.
But little did he know that his neighbor, who happened to be a math genius, had been observing him from afar. The neighbor, with a sly grin on his face, approached the farmer and said, "Hey buddy, I hate to break it to you, but you could have actually gotten a larger field!"
The farmer was puzzled and asked, "How is that possible?"
The neighbor explained, "Well, you see, in order to maximize the area, you would need to divide the length of fencing into equal parts for the length and width. So if you divide it into 800 feet for each, you would actually have a rectangular field with dimensions of 800 feet by 800 feet. That would give you a larger area!"
The farmer scratched his head in disbelief, but then he realized that his neighbor was right. He thanked the neighbor for enlightening him and went on to create an even bigger and better field. And they all lived happily ever after.
So, according to this humorous scenario, the dimensions of the largest possible field would be 800 feet by 800 feet. But please remember to do the actual calculations to get the precise answer!
To find the dimensions of the field with the largest area, we can set up an equation using the given information.
Let's denote the length of the rectangular field as L and the width as W.
From the problem statement, we know that the farmer has a total of 2400 ft of fencing. Since the field borders a straight river, the length of the fence required will be the sum of the lengths of the three sides that are not adjacent to the river:
L + W + L = 2400
Simplifying the equation, we have:
2L + W = 2400
Now, we need to express the area of the field, A, in terms of L and W. The area of a rectangle is given by:
A = L * W
To solve for the dimensions of the largest possible field, we need to find the maximum value of A. This can be done by finding the critical points of A, which occur when the derivative of A with respect to either L or W is equal to zero.
Differentiating A with respect to W:
dA/dW = L
Setting dA/dW = 0, we find that L = 0.
Differentiating A with respect to L:
dA/dL = W
Setting dA/dL = 0, we find that W = 0.
Since L and W cannot be zero in this context, we conclude that the dimensions of the largest possible field occur at the points where dA/dW = 0 and dA/dL = 0.
Substituting L = 0 into the equation 2L + W = 2400, we find that W = 2400.
Substituting W = 0 into the equation 2L + W = 2400, we find that L = 1200.
Therefore, the dimensions of the field with the largest area that can be fenced off are L = 1200 ft and W = 2400 ft.
To find the dimensions of the rectangular field with the largest possible area, we can follow these steps:
1. Let's start by drawing a diagram to visualize the situation. The river will be represented as a straight line, and the rectangular field will be on one side of the river.
2. Since the river is straight, we know that the length of the field must be perpendicular to the river. Let's call this length x and represent it as the base of the rectangle.
3. The remaining 3 sides of the rectangle will be parallel to the river and have equal lengths. Let's call this length y and represent it as the two sides adjacent to the river.
4. According to the problem, the farmer has 2400 ft of fencing. We can determine the perimeter of the rectangle using the given dimensions x and y: 2y + x + x = 2400.
5. Simplifying the equation, we have 2x + 2y = 2400. Rearranging this equation gives us y = 1200 - x.
6. To find the area of the rectangular field, we multiply the length (x) by the width (y). A = x * y.
7. Substituting the value of y from step 5 into the area equation gives us A = x * (1200 - x).
8. We can expand the equation to get A = 1200x - x^2.
9. The area of the rectangle is a quadratic equation, and the maximum area occurs when its vertex is the highest point on the parabola. In this case, the vertex represents the largest possible area.
10. To find the x-coordinate of the vertex, we can use the formula x = -b / (2a), where a = -1 and b = 1200. Plugging in the values, x = -1200 / (2*(-1)) = 600.
11. We have found the x-coordinate of the vertex, which is 600. To find the corresponding y-coordinate, we can substitute x = 600 into the equation from step 7: A = 600 * (1200 - 600).
12. Simplifying, A = 600 * 600 = 360,000 sq ft.
13. Therefore, the dimensions of the field with the largest possible area are x = 600 ft and y = 1200 - x = 600 ft.
So, the dimensions of the field with the largest possible area that the farmer can fence off are 600 ft by 600 ft.
Length of side parallel to river, (we only have one such side) ---- y ft
width of rectangle --- x ft
so 2x + y = 2400
or y = 2400 - 2x
area = xy = x(2400 - 2x)
= -2x^2 + 2400x
this is represented by a parabola which opens downwards,
the x of the vertex is -2400/-4 = 600
then y = 2400 - 2(600) = 1200 ft