Question

A rancher wants to fence in an area of 1000000 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

The shortest length of the fence used by the rancher in a rectangular field of 1000000 square feet is 4899 ft.

What is the area of a rectangle?

The area of a rectangle is a region enclosed in four corners where two opposite sides are equal. The area of the rectangle is mathematically expressed as:

Area of a rectangle = length(x) × breath(y)

1000000 sq.ft = (x) × (y) ---- (1)

However, to determine the fencing, we have:

Length of fencing = 2x + 2y  

• since the rectangle has four sides and opposite sides are equal

Suppose we use the x-direction to estimate the region where the fencing is located, we have:

Length of fencing = 2x + 2y + x

Length of fencing = 3x + 2y ---- (2)

From equation (1), make (y) the subject of the formula and replace the value of y with equation (2)

The Length of fencing is:

$\mathbf{L = 3x - 2( \dfrac{1000000}{x})}$

$\mathbf{\dfrac{dL}{dx} = 3 - \dfrac{2000000}{x^2} = 0}$

$\mathbf{ x^2= \dfrac{2000000}{3} }$

$\mathbf{ x= \dfrac{1000\sqrt{2}}{\sqrt{3}} }$

$\mathbf{x = \dfrac{1000 \sqrt{6}}{3}}$

Now, if we replace the value with equation (1), then y will be:

$\mathbf{y = \dfrac{1000000}{\dfrac{1000 \sqrt{6}}{3}} }\\ \\ \\ \\ \mathbf{y = \dfrac{3000 }{\sqrt{6}}}$

$\mathbf{x =500\sqrt{6}}$

Now, the shortest length of the fencing can be computed by using the equation (2):

L = 3x + 2y

$\mathbf{L = 3( \dfrac{1000 \sqrt{6}}{3}) + 2(500 \sqrt{6})}$

$\mathbf{L =1000 \sqrt{6} + 1000 \sqrt{6}}$

$\mathbf{L =2000 \sqrt{6} }$

$\mathbf{L \simeq 4899 \ ft}$

Learn more about the area of a rectangle here:

brainly.com/question/25292087