Question

g Farmer Violet wants to enclose one 990 square foot rectangular area of land using fencing. One side will use special fencing costing $12 per foot. The remaining three sides of fencing will be formed using standard fencing costing $10 per foot. Find the dimensions of the corral that would minimize the cost of the fencing required for this project, rounded (if necessary) to the nearest hundredth of a foot

Answer 1

Answer:

Sides of the rectangular area:

x =  30  f

y =  33  f

Step-by-step explanation:

Let´s call   x  and y the sides of the rectangular area

A = x*y  = 990          y  = 990/x

The function cost is:

C = Costs ( sides x)  + costs ( sides y )

Cost of sides x    =  12*x +  10x

Cost of sides y    =  10*y  +  10 *y   =  2*10*y  = 20*y

C = 12*x + 10*x + 20*y =  22*x  + 20*y

The function cost as function of x is:

C(x)  = 22*x + 20*(990/y)  =  22*x  +  19800/x

Tacking derivative on both sides of the equation:

C´(x)  = 22  + [ - 19800/x²]

C´(x) = 0       22  - 19800/x²  =  0

Solving for x

22*x²  - 19800  =  0

22x²  =  19800

x² =  19800/22 =  900

x₁,₂ = ± 30        We dismiss negative root ( we never have negative lenghts)

Then     x =  30 f

And  y  =  990 / 30      y  = 33  

To see if x = 30 is a minimum for function C(x) we evaluate the second derivative.

C´´(x) = 2*x* 19800/x⁴      

C´´ (x) = 39600/x³    so C´´ is always greater than 0 then C (x) has a minimum at x = 30

Sides of the rectangular area are:

x  =  30

y  =  33