SOLUTION: A rectangular lot is 65 yards long and 110 yards wide. Give the length and width of another rectangular lot that has the same perimeter but a smaller area.

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Question 1206101: A rectangular lot is 65 yards long and 110
yards wide.
Give the length and width of another rectangular lot that has the same perimeter but a smaller area.
Found 3 solutions by mananth, greenestamps, ikleyn:
Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!
A rectangular lot is 65 yards long and 110
yards wide.
Give the length and width of another rectangular lot that has the same perimeter but a smaller area.
perimeter of the given rectangle 2*(65+110)=350
Area = 110*65=7150
2l+2w=350
l+w=175
there can be many possibilities for L & B
Square is a rectangle with all sides equal and at 90 degree to each other
For a given perimeter, square has the maximum area .









Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The sum of the length and width of the given rectangle is 175 yards.

For another rectangle to have the same perimeter, the sum of its length and width has to be 175 yards also.

The area of a rectangle is the product of the length and width.

For a fixed sum of two numbers, the product of the two numbers is larger if the two numbers are close together and smaller if the two numbers are farther apart. The product is maximum if the two numbers are the same.

So the rectangle with the largest area and with the same perimeter as the given rectangle is a square with side length 175/2 = 87.5.

Some other dimensions of a rectangle with the same perimeter as and larger area than the given rectangle are...
70 x 105
75 x 100
87 x 88
...

Or, in general, any two numbers of the form (87.5+x) and (87.5-x), where x is any non-negative number less than 87.5.
Answer by ikleyn(52805)   (Show Source): You can put this solution on YOUR website!
.
A rectangular lot is 65 yards long and 110 yards wide.
Give the length and width of another rectangular lot that has the same perimeter but a smaller area.
~~~~~~~~~~~~~~~~~~~~~


        In the post by @mananth, there is  NO  ANSWER  to the problem's question.
        So,  I came to bring a direct explicit and complete answer. 


The given rectangle has the perimeter of  2*(65+110) = 130 + 220 = 350 yards
and the area of 65*110 = 7150 square yards.


To get another rectangle with the same perimeter and smaller area, 
consider a more narrow rectangle with one dimension 65-x and other dimension 110+x,
where x is any positive value (less than 65).


For example, let's take x = 5 yards.
Then the dimensions of the new rectangle are 65-5 = 60 yards and 110+5 = 115 yards.

Then the new perimeter is 2*(60+115) = 120 + 130 = 350 yards  (the same as the original rectangle has)
and the area 60*115 = 6900 square yards (which is less than the area of the original rectangle).


So, the answer is just given (one of many other possible answers).


To get the formal proof, notice that dimensions 65-x and 110+x provide the same perimeter
(which is obvious), but smaller area, because

    65*110 - (65-x)*(110+x) = 65*110 - 65*110 + 110x - 65x + x^2 = (110-65)x + x^2 = 45x + x^2,

is positive for any positive x.

Solved, answered and fully explained.



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