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

Sides of the rectangular area:

x = 30 f

y = 33 f

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