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
