# Question #3e1d2

##### 2 Answers

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 = (2 #xx#Length) + (2#xx#Width)*

**But:**

We know that the maximum perimeter must be 3600 yards as this is the maximum length of fence that we've got.

*Perimeter**(2 #xx#Length) + (2#xx#Width)*

**Rearranging linearly gives:**

*(2*#xx# Length)=

*(2*#xx# Width)

**Dividing both sides by 2 gives:**

*Length*=

*Width*

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

*Area* *Length**Length* )

**Now we'll let Length = #L# and Area = #A# for simplicity:**

*A(L)*

*L*

*L*)

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

If we now derive this expression with respect to

Consider for a polynomial of any power:

If

Thus:

**A maximum exists where #A'(L)=0# **

Thus for maximum area we must have a length of 900 yards.

Our width would *Width* from above.

Thus width would also be 900 yards.

**NB: Notice how this forms a square?**

The maximum area produced would

*Area*=*yards squared.*

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

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 , 2

You have found the relation between

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 =

Like stated before , you have to take the derivative of A either with respect to length or breadth . We have both

we had relation of

I have chosen to substitute for

now,

we can take derivative of A wrt l.

When you have reached the maximum or minimum

so we can go ahead and say for the maximum area

=>

=>

So we got length as 900 , so we can find

So the length is 900 , breadth is 900.

Now , as said before , when you say

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 .