Mathematics: A farmer wants to build a rectangular pen and then divide it with two interior fences. The total area inside of the pen will be 264 square meters. The exterior fencing costs $15.60 per meter and the interior fencing costs $13.00 per meter. Find the dimensions of the pen that will minimize the cost.

Area = base x height /2Base =26.5Height =4026.5 x 40 /2 = 530...

A farmer wants to build a rectangular pen and

trin83

11.08.2021

x = 12 m and y = 22 m

Step-by-step explanation:

Total area = 264 m^2

∴ xy = 264

$y=\frac{264}{x}$ ............(1)

Cost function = C(x,y) = 2 x (15.60) + 2y(15.60) + 2x(13)

C(x,y) = 57.2 x + 31.2y

Therefore, using (1),

$C(x) = 57.2x+31.2 \left(\frac{264}{x} \right)$

$C(x) = 57.2x+\frac{8236.8}{x} \right)$

So, cost C(x) minimum where C'(x) = 0

$C'(x) = 57.2 - \frac{8236.8}{x^2}=0$

$x^2=\frac{8236.8}{57.2}$

$x^2=144$

$x=12$ m

Therefore, $y=\frac{264}{x}$

$=\frac{264}{12}$

= 22 m

So the dimensions are x = 12 m and y = 22 m.

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