Question 39858
1. The total profit by selling 'x' boxes of candies is 
P = $(x(5-0.05x) - 1.5x) = ${{{(3.5x - 0.05x^2)}}}
For maximizing 'P', {{{dP/dx = 0}}} and {{{d^2P/dx^2 < 0}}}
Now, {{{dp/dx = 0 = 3.5 - 2*0.05*x = 3.5 - 0.1x}}}
or x = 35
and {{{d^2p/dx^2 =  - 2*0.05 = -0.1}}}
Thus P is maximum for x = 35 and corresponding P = ${{{(3.5*35 - 0.05*35^2)}}} = $61.25.
So for maximum profit, 35 boxes are to be sold and the maximum profit is $61.25.


2. Perimeter = 3000 ft.
Let length = L ft, width = W ft of the rectangular field.
Then, 2(L + W) = 3000 or L + W = 1500 _________(1)
Given: L = x, then from (1) W = 1500 - x.
Hence, area A = {{{L*W = x*(1500-x)}}} sq ft
To find maximum area, maximize A w.r.t x as done in the problem above.
Then you get, x = 750 for maximum A and this maximum value of A is {{{750*750}}} = 562500
Hence area of the rectangular field is maximum when its each side is 750 ft i.e. the field is a square with side 750 ft and this maximum area enclosed is 562500 sq ft.


[V.V.I.: From this problem we come to the conclusion that of all rectangles with same perimeter, the square has the largest area.]