Arectangularplotoflandistobefencedinusingtwokinds of fencing. Two opposite sides will use heavy-duty fencing selling for $3 a foot, while the remaining two sides will use standard fencing selling for $2 a foot. What are the di- mensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000?

Answer:Dimensions of the rectangular plot will be 500 ft by 750 ft.Step-by-step explanation:Let the length of the rectangular plot = x ft.and the width of the plot = y ft.Cost to fence the length at the cost $3.00 per feet = 3xCost to fence the width of the cost $2.00 per feet = 2yTotal cost to fence all sides of rectangular plot = 2(3x + 2y)2(3x + 2y) = 6,0003x + 2y = 3,000 ----------(1)3x + 2y = 3000       2y = 3000 - 3x       y = [tex]\frac{1}{2}[3000-3x][/tex]       y = 1500 - [tex]\frac{3x}{2}[/tex]Now area of the rectangle A = xy square feetA = x[[tex]1500-\frac{3x}{2}[/tex]]For maximum area [tex]\frac{dA}{dx}=0[/tex]A' = [tex]\frac{d}{dx}(1500x-\frac{3}{2}x^{2})[/tex] = 01500 - 3x = 03x = 1500x = 500 ftFrom equation (1),y = 1500 - [tex]\frac{3}{2}\times 500[/tex]y = 1500 - 750y = 750 ftTherefore, for the maximum area of the rectangular plot will be 500 ft × 750 ft.two fencing 3(500+500) = $3000other two fencing 2(750+750) = $3000