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?
500x250
@ Steve I don't completely understand I want to knowhow we got to that answer/
well, draw a diagram. If the width is x, then the length is 1000-2x. So, the area is
a = x(1000-2x) = 1000x-2x^2
da/dx = 1000-4x
da/dx=0 when x=250
As is usual, you will find that for maximum area, the fencing is divided equally among the widths and lengths. In this case, that means 500 ft each.
2 widths of 250' each, and 1 length of 500.
Naturally, for a 4-sided enclosure, that just means a square.
Ok thanks I don't quite get it but I will follow your steps and Draw the diagram :)
To find the maximum area that can be enclosed with 1000m of fencing, we need to determine the dimensions of the rectangular storage area.
Let's assume the length of the rectangular storage area is "L" and the width is "W". There are two lengths and two widths, so the total fencing required would be: 2L + 2W.
We also know that there are three sides where a security fence is required. Therefore, the remaining side without a fence is equal to the width W.
Since the total fencing required is equal to 1000m, we can create an equation:
2L + 2W + W = 1000
Simplifying the equation, we get:
2L + 3W = 1000
Now, let's solve for W in terms of L:
3W = 1000 - 2L
W = (1000 - 2L) / 3
The area of the rectangular storage area (A) is equal to the length multiplied by the width: A = L * W
Substituting the value of W from above:
A = L * [(1000 - 2L) / 3]
To find the maximum area, we need to maximize the value of A. We can do this by taking the derivative of A with respect to L, setting it equal to 0, and solving for L.
dA/dL = (1000 - 4L) / 3
Setting dA/dL = 0:
1000 - 4L = 0
4L = 1000
L = 1000 / 4
L = 250
Substituting the value of L back into the equation for the width:
W = (1000 - 2*250) / 3
W = 250
Therefore, the dimensions of the rectangular storage area that would enclose the maximum area with 1000m of fencing are: L = 250m and W = 250m.
Finally, we can calculate the maximum area by substituting the values of L and W into the equation for A:
A = 250 * 250
A = 62500 square meters
So, the maximum area that can be enclosed with 1000m of fencing is 62500 square meters.