I need step by step work. a. you have $2000 dollars to buy fence to enclose a re

Question

I need step by step work.

a. you have $2000 dollars to buy fence to enclose a rectangular plot of land. The fence for the top and bottom costs $5 per foot and for the sides it costs $2 per foot. Find the dimensions, x and y, of the plot with the largest area. For this largest plot, how much money was used for the top and bottom, x, and for the sides, y?

b. you have G dollars to buy fence to enclose a rectangular plot of land. The fence for the top and bottom costs $5 per foot and for the sides it costs $2 per foot. Find dimensions of the plot with the largest area. For this largest plot, how much money was used for the top and bottom,x,and for the sides, y?

c. you have H dollars to buy fence to enclose a rectangular plot of land. The fence for the top and bottom costs $A per foot and for the sides it costs $B per foot. Find dimensions of the plot with the largest area. For this largest plot, how much money was used for the top and bottom, x, and for the sides, y?

Explanation / Answer

a) 5*(x+x)+2*(y+y)=2000 is the cost function

x*y is the area we want to maximize

10x+4y=2000 so x=200-4/10*y

so we maximize (200-4/10*y)*y=200y-4/10*y^2

the derivative is 200-4/5*y so 200=4/5*y so y=250 and then x=100.

b) 5*(x+x)+2*(y+y)=G is the cost function

x*y is the area we want to maximize

10x+4y=G so x=G/10-4/10*y

so we maximize (G/10-4/10*y)*y=G/10y-4/10*y^2

the derivative is G/10-4/5*y so G/10=4/5*y so y=G/8 and then x=G/10-G/20=G/20.

c) A*(x+x)+B*(y+y)=H is the cost function

x*y is the area we want to maximize

2Ax+2By=H so x=H/2A-B/A*y

so we maximize (H/2A-B/A*y)*y=H/2A*y-B/A*y^2

the derivative is H/2A-2B/A*y so H/2A=2B/A*y so y=H/4B and then x=H/2A-B/A*H/4B=H/4A

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