An unfenced rectangular plot of land with an area of 15000 square feet is to be split up into two rectangular gardens. Find the minimum amount of fencing required to define and separate these two gardens? Where do I begin? I just want to know how to set up the equation to figure out the problem.
you ant two rectangles with one side in common. So, if the big plot has dimensions x and y, then
xy = 15000
Now, there will be two sides of length x and 3 sides of length y/3 (because of the extra fence down the middle), so the amount of fence is
f = 2x+3y = 2x + 45000/x
Now just find where df/dx is zero, and that will give you the dimensions for minimum fence.
Expect that when you are all done, the fencing will be divided among the lengths and widths. That is, you will see that 2x=3y.
To find the minimum amount of fencing required, you need to determine the dimensions of the rectangular gardens. Let's denote the dimensions of the two gardens as length1, width1, length2, and width2.
Since the total area of the plot is given as 15000 square feet, we can set up the following equation:
length1 * width1 + length2 * width2 = 15000
Now, since the two rectangular gardens need to be separated, we know that the combined length of the two gardens (length1 + length2) needs to be equal to the total length of the plot. Similarly, the combined width of the two gardens (width1 + width2) must be equal to the total width of the plot.
Therefore, we can set up the following equations:
length1 + length2 = total length of the plot
width1 + width2 = total width of the plot
Now, we need to consider the fencing required for the boundaries of the two gardens.
The length of the fencing required for the boundaries of garden 1 is: 2 * (length1 + width1)
The length of the fencing required for the boundaries of garden 2 is: 2 * (length2 + width2)
Hence, the total length of fencing required to define and separate the two gardens is:
2 * (length1 + width1) + 2 * (length2 + width2)
To minimize the amount of fencing required, we need to find the values of length1, width1, length2, and width2 that satisfy these equations and minimize the total length of fencing required.