Question 42205
Let length = L ft, width = W ft.
Then perimeter = 300 = 2(L + W)
or L + W =150 _____(1)


Now, area is given by
A = W x L 
= (150 - L) x L 
= {{{150L - L^2}}}
= {{{75^2 - (L^2 - 2*L*75 + 75^2)}}}
or {{{A = 75^2 - (L - 75)^2}}}________(2)


For 'A' to be maximum, {{{(L - 75)^2}}} has to be minimum.
But it is a real square and hence cannot be negative.
So its minimum value is zero. 
So, {{{(L - 75)^2 = 0}}}
or L = 75


We have obtained L = 75. Putting this value in (1) we find W = 75.


Putting L = 75 in (2) we will get the maximum value of 'A'.
A{{{(max) = 75^2 - (75 - 75)^2 = 75^2}}} = 5625


So the maximum area of the patio is 5625 sq ft and its dimensions are 75 ft x 75 ft. In other words for maximum area, the patio is a square with side 75 ft.