Question #3e1d2

2 Answers
Dec 14, 2017

Length and width of 900 yards.

Explanation:

Consider that the area of a rectangle can be evaluated as:
Area = Length xx Width

Also consider that the total perimeter of the rectangle must be:
Perimeter = (2xxLength) + (2xxWidth)

But:
We know that the maximum perimeter must be 3600 yards as this is the maximum length of fence that we've got.
:. Perimeter=3600=(2xxLength) + (2xxWidth)
Rearranging linearly gives:
=>3600-(2xxLength) =(2xxWidth)
Dividing both sides by 2 gives:
:. 1800-Length = Width

Consider that this value can now be used within the area equation, allowing for the elimination of one variable.

Area = Lengthxx (1800-Length )
Now we'll let Length = L and Area = A for simplicity:
:. A(L) = Lxx (1800-L )
A(L) = 1800L - L^2

This gives us an expression for area in terms of one variable only!

If we now derive this expression with respect to L we can evaluate when the function is maximal:

Consider for a polynomial of any power:
If f(x)=ax^n
f'(x)=nax^(n-1)

Thus: A'(L) = 1800 - 2L^1
A maximum exists where A'(L)=0

:. 0=1800-2L
2L=1800
L=900
Thus for maximum area we must have a length of 900 yards.

Our width would :. be 1800-900#=Width from above.
Thus width would also be 900 yards.
NB: Notice how this forms a square?
The maximum area produced would :. be:
Area=900xx900=810,000 yards squared.

Dec 14, 2017

length = 900 yards.
breadth = 900 yards.

Explanation:

mona has 3600 yards of fence. She wants the maximum area. So ,she would naturally use all of her fencing.
The fence has to be made a rectangle having a length of l units and a breadth of b units.
The length of fence is indeed the perimeter of the rectangle she wants.
So perimeter of the rectangle is 2(l+b) units ,and that is equal to 3600 yards.
So , 2l + 2b = 3600. ---> let this be equation 1.

You have found the relation between l and b. Wherever you want , you can express l in terms of b and vice versa.

Now we want the maximum area possible with perimeter of 3600 yards.

The easier way to maximize/minimize a function(remember ,area is a function of length and breadth) is to express it as derivative with respect to any one of its parameters( like , length or breadth).

area of the rectangle is
A = l * b.
Like stated before , you have to take the derivative of A either with respect to length or breadth . We have both l and b here . So what can we do?
we had relation of l and b in equation 1.

I have chosen to substitute for b as (3600 - 2l)/2 . (you can always substitute the other way. You just need either this or that in the expression , which makes things easier)

now, A = l * (3600 - 2l)/2 or ,
A = (3600 l - 2l^2)/2
A = 1800 l - l^2

we can take derivative of A wrt l.

(dA)/(dl) = 1800 - 2l ---> Eq 2. We will need this again at the end.

When you have reached the maximum or minimum (dA)/(dl) or any derivative for that matter , it becomes zero.

so we can go ahead and say for the maximum area
(dA)/(dl) = 1800 - 2l = 0
=> 1800 = 2l
=> 900 = l .
So we got length as 900 , so we can find b
b = (3600 -2l)/2 = 1800 - l = 900
So the length is 900 , breadth is 900.

Now , as said before , when you say (dA)/(dl) is zero , it means it has a chance to be maximum or minimum. So how do you know that you have got the maximum area?

take derivative of Eq 2. wrt to l.
if you get a negative value it means area is maximum . if you get a positive value area is minimum .