Question

If the farmer has 234 feet of fencing, what are the dimensions of the region which enclose the maximal area?

Answer 1

Answer: 58 ft × 58 ft  

Step-by-step explanation:

Let the length of the region = x feet

And, the width of the region = y feet

Since, the perimeter of the region = 234 feet ( Given )

⇒ $2(x+y) = 234$

⇒ $x+y = 117$

⇒ $y = 117 - x$

Again the area of the region, A = xy

⇒ $A(x) = x(117-x)$

⇒ $A(x) = 117x - x^2$

By differentiating the above equation with respect to x,

⇒ $A'(x) = 117 - 2x$  

For maxima or minima,

$A'(x) = 0$

$\implies 117 - 2x = 0$

$\implies -2x = -117$

$\implies x = 58.5$

Again differentiating equation A'(x) with respect to x,

We get, A''(x) = -2

Hence, For x = 58.5 A''(x) = negative

⇒ For x = 58.5 feet the area A(x) is maximum,

⇒ The length of the region having maximum area = x = 58.5 feet

And, the width of the region having maximum area = y = 117-x= 117 - 58.5=58.5 feet,

⇒ The dimension of the region having the maximum area = 58.5 ft × 58.5 ft

Answer 2

Answer:

Step-by-step explanation:

Lets suppose the area to be enclosed is a square and let the side be s.

The perimeter would be 234 feet.

Perimeter of square is = 4 x side

We get;

$4s=234$

Dividing both sides by 4;

s = 58.5 feet

Now, area of the square = side x side

So, the maximum area that can be enclosed is = $58.5\times58.5=3422.25$ square feet

Hence, the dimensions will be 58.5 feet by 58.5 feet.