1000m of fencing is to be used to make a rectangular enclosure,Find the greatest possible area,and the corresponding dimensions if the length of the enclosure is xm and the width is (500-x)m.
let the lenght of the rectangle be x meters.
Then, the witdh is (1000-2x)/2 meters. So the width will be 500-x meters.
The area A of the rectangle is x*(500-x) da/dx of 500x-x^2 is 500-2x which is equal to 0.
Make x the subject and get x = 250.
Area is maximum when x=250 m.
Area=250*(500-250) which is equal to 62500m^2.
as with all these problems, the maximum area is attained when the fencing is divided equally between lengths and widths. In this case, since there are no missing sides or internal divisions, that means a square.
So, with 1000m of fencing, the maximum area is when the pen is a square 250m on a side.
maximum area is 250^2 m^2
250
To find the greatest possible area of the rectangular enclosure, we need to maximize the area function with respect to the given constraints.
Let's start by writing the formula for the perimeter of the rectangular enclosure:
Perimeter = 2(length + width)
We are given that the total length of fencing is 1000m. Therefore, we can write the equation:
1000 = 2(x + (500 - x))
Simplifying this equation gives:
1000 = 2(500)
Next, let's write the formula for the area of the rectangular enclosure:
Area = length * width
Substituting the value of width in terms of x, we get:
Area = x * (500 - x)
Now, we have the area function in terms of x, which is the variable we're going to optimize to find the maximum area.
To find the maximum area, we'll take the derivative of the area function with respect to x and set it equal to 0. This will give us the value of x that maximizes the area.
d(Area)/dx = 500 - 2x
Setting this equal to 0, we solve for x:
500 - 2x = 0
2x = 500
x = 250
Therefore, the maximum area is achieved when x = 250. Substituting this value back into the equation for width, we find:
Width = 500 - x
Width = 500 - 250
Width = 250m
So, the corresponding dimensions for the greatest possible area are: length = 250m and width = 250m.
Finally, we can calculate the maximum area by substituting the values of x and width into the area formula:
Area = length * width
Area = 250m * 250m
Area = 62500 square meters
Hence, the greatest possible area of the rectangular enclosure is 62500 square meters, and the corresponding dimensions are length = 250m and width = 250m.