A rectangular field

Question

A rectangular field is enclosed by fencing worth a total of $60000. Due to strong south winds, the fence on the north and south sides cost $3 per meter, while the fence on the east and west side coat only 2$ per meter. A) write the formula for the area of the rectangle in terms of x and y. B) write an equation for the total cost of the fence in terms of x and y. C) find a formula for the area a as a function of x. D) find the value of x which maximized the value of the area A. What point dies this represent on the function you found above? E) state the dimensions (x and Y) which maximize the fields area.

 

Answer 1

The cost of the fence is

 

3$/meter(2x)+2$/meter(2y)

=$60000

 

Solve for y in terms of x

 

y=(60000-6x)/4

y=15000-3x/2

 

The area of the enclosed rectangle is

a=xy

 

Subtitute y for the above function of x to get the area in terms of x only

 

a=x[15000-3x/2]

 

a=-3/2(x²)+15000x

 

To maximize the area take the derivative. Set it to zero. And then solve for   

 

a'=-3x+15000=0

x=5000 meters

 

to establish that this is a maximum not a minimum get the second derivative 

 

a''=-3 which is negative so any extrema is a maximum 

 

remember

y=15000-3x/2

 

to solve for y

y=15000-3(5000)/2

y=7500 meters 

 

check

 

3$/meter(2•5000 meters)

+2$/meter(2•7500 meters)
=$60000

 

$30000+$30000=$60000

 

Checks!