A farmer uses 1034 meters of fencing to enclose a rectangular region and also to subdivide the region into three smaller rectangular regions by placing a fence parallel to one of the sides. Find the demensions that produce the greatest enclosed area.
After making your sketch and creating the 3 rectangles
let the length of each rectangle be x, let the width of each rectangle be y
So the area of the whole enclosed area = 3xy
also : 4x + 6y = 1034
6y = 1034 - 4x
y = (1034 - 4x)/6
area = 3xy - 3x(1034-4x)/6
= 517x - 2x^2
If you know Calculus ....
d(area)/dx = 517 - 4x
= 0 for a max of area
x = 517/4 =129.25
y = 86.167 m
(maximum area = 3xy = 33411.125
Check by taking a value of x slightly smaller and slightly larger and finding their areas
e.g
x =129 , then y = 86.3333
area = 3xy = 33411 , just a bit smaller
x = 130, then y = 85.6666..
area = 3xy = 33410 , again a bit smaller
my answer is plausible.
What is the answer for 104.19 + 4.9
To find the dimensions that produce the greatest enclosed area, we need to maximize the area function. Let's represent the length of the rectangular region as 'l' and the width as 'w'.
Given that the perimeter of the rectangular region is 1034 meters, we can write the equation:
2l + 3w = 1034
To solve for 'l' in terms of 'w', we isolate 'l':
2l = 1034 - 3w
l = (1034 - 3w) / 2
Now, we can express the area of the rectangular region, A, as the product of its length and width:
A = l * w
Substituting the value of 'l' from the previous equation, we have:
A = [(1034 - 3w) / 2] * w
Next, we need to find the value of 'w' that maximizes the area function 'A'. To do this, we can take the derivative of 'A' with respect to 'w' and set it equal to zero, as follows:
(dA/dw) = 0
We can then solve this equation to find the critical value(s) of 'w'. To simplify calculations, let's multiply through by 2:
d/dw [(1034 - 3w)w] = 0
Expanding and differentiating, we get:
1034w - 3w^2 = 0
Next, we can factor out 'w':
w(1034 - 3w) = 0
This equation gives two critical values for 'w': w = 0 and w = 1034/3.
However, the width of the rectangle cannot be zero, so we discard w = 0.
Therefore, the only critical value of 'w' is w = 1034/3.
Now, we substitute this value back into the equation for 'l' to find the corresponding value of 'l':
l = (1034 - 3(1034/3)) / 2
l = 1034/2 - 1034/2
l = 517 - 517
l = 0
We find that when w = 1034/3, the corresponding value of l is 0. However, a rectangle with a width of 1034/3 and a length of 0 does not make sense.
Since we cannot have a rectangle with one side equal to zero, the only option is to assume the problem is impossible to solve.