a rectangular storage area is to be constructed along the sides of a tall building. A security fence is required along the three remaining sides of the area. What is the maximum area that can be enclosed with 1000m fencing?

Question

a rectangular storage area is to be constructed along the sides of a tall building. A security fence is required along the three remaining sides of the area. What is the maximum area that can be enclosed with 1000m fencing?

Answer 1

so one side is not needed.

let the two width be x m
let the one length be y m

2x + y = 1000
y = 1000 - 2x

area = xy
= x(1000-2x)
=1000x - 2x^2

d(area)/dx = 1000 - 4x
= 0 for a max of area
4x = 1000
x = 250
y = 1000-500 = 500

max area = xy
= 250(500)
= 125000 m^2

Answer 2

Reiny you are a Godsent Thanks Alot ^_^

Feeling more confident for finals now!

Answer 3

so is it a problem if I used dA/dx instead?

Answer 4

Ah, the joys of fencing! Let's see if I can solve this one for you.

To maximize the area, we want to create a square shape since it has the largest area for a given perimeter. So, let's break it down!

Since there are three sides of the storage area already adjacent to the building, we only need to focus on constructing the fourth side with the fencing.

Let's say the length of the fourth side is x meters. Since we want to enclose the maximum area, the length and width of our square storage area will both be x.

The perimeter of this square storage area is 3x, which means we have 1000m of fencing to work with. So, we can set up the equation:

3x = 1000

Now, solving for x:

x = 1000/3

Great! We found the length of the fourth side, which is approximately 333.33 meters.

Now, to find the maximum area, we square the length:

Area = x * x = (1000/3) * (1000/3) = 111,111.11 square meters

So, the maximum area that can be enclosed with 1000m of fencing is approximately 111,111.11 square meters.

Now, that's what I call a square deal!

Answer 5

To find the maximum area that can be enclosed with 1000m of fencing along the sides of a tall building, we need to determine the dimensions of the rectangular storage area.

Let's assume the length of the storage area is L and the width is W. Since there are three sides requiring fencing (two widths and one length), we can express the perimeter of the storage area as:

2L + W = 1000m (Equation 1)

Also, the area enclosed by the storage area is given by:

Area = L * W (Equation 2)

To find the maximum area, we need to maximize Equation 2 subject to Equation 1.

To do that, we can solve Equation 1 for L:

2L = 1000m - W
L = (1000m - W)/2

Now, substitute this value of L in Equation 2:

Area = [(1000m - W)/2] * W

Taking the derivative of Area with respect to W and setting it equal to zero will give us the maximum area:

d(Area)/dW = (1000m - 2W)/2 - W = 0
1000m - 2W - 2W = 0
1000m = 4W
W = 1000m/4
W = 250m

Now that we have the width, we can substitute this value back into Equation 1 to find the length:

2L + W = 1000m
2L + 250m = 1000m
2L = 1000m - 250m
2L = 750m
L = 750m/2
L = 375m

Therefore, the dimensions of the rectangular storage area that will enclose the maximum area with 1000m of fencing are:
Length = 375m
Width = 250m

To find the maximum area, substitute these dimensions into Equation 2:

Area = L * W
Area = 375m * 250m
Area = 93750m^2

So, the maximum area that can be enclosed with 1000m of fencing is 93750 square meters.