Question

rancher wants to fence in an area of 2000000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use

Answer 1

Answer:

• 7000ft

Step-by-step explanation:

let's break down 2,000,000 into its multiples,

Multiples of 2,000,000= 2⁷ × 5⁶

Using above values to find different combinations of length(l) and breadth (b) of rectangle and corresponding parameter of rectangle

• l=5,   b=400,000  ,parameter= 2(5) + 2(400,000)= 800,010, total length of fence required= parameter+ side with shortest length= 800,015

• l=2   b=1,000,000 parameter= 2,000,004, shortest fence required= 2,000,008

• l=4,  b= 500,000 parameter= 1,000,008, shortest fence required= 1,000,012
• From above, we can see a trend that the parameter of rectangle decreases if length and breadth are increased provided that area is constant. So, a rectangle will have shortest parameter if all of its sides are equal.
• length of side of rectangle with shortest possible parameter= 2^3 ×5^3= 1000 and breadth of side of rectangle with shortest possible parameter= 2^4× 5^3=2000
• Shortest possible length of fence= 2(2000)+2(1000)+1000=7000ft

Answer 2

Answer:

The shortest length of the fence is 6928.203 ft.

Step-by-step explanation:

We can define the area of the field (A) as a function with constant value,

A=xy=2000000 ft^2

And, in the same way, we can define one of the side in terms of the other side,

y=2000000/x

On the other hand, we can define the length of the fence (L) as the perimeter of the field (P) plus the length of one of the sides:

L=P+x

L=(2x+2y)+x

L=3x+2y

But, above we defined y in terms of x, so we need to replace it in the equation that defines L:

L=3x+2(2000000/x)

L=3x+4000000/x

The problem asks us about the shortest length of fence, it means the minimum value of L. Then, we need to find the minimum value of L, and we can do this, if we derivate the function L and equalize the derivate of the function to zero,

$\frac{dL}{dx} =0$

$\frac{dL}{dx} =3-\frac{4000000}{x^{2} } =0$

$3=\frac{40000000}{x^{2} }$

$x^{2} =\frac{4000000}{3}$

And, we know that a square root has two valids results: one negative and one positive, so,

$x=±\sqrt{\frac{4000000}{3} }$

In this case, a lenght cannot be negative, so we are going to take the positive value,

$x=\sqrt{\frac{4000000}{3} } =1154.700$

So, taking into account the function that defines L, the shortest lenght for the fence will be:

$L=3(1154.700)+\frac{4000000}{1154.700}$

$L=6928.203 ft$