a rancher wants to fence in an area of 500,000 square feet in a rectangular field and then divide it in half…

Explanation:

Step1: Define variables

Let the length of the field be $x$ and the width be $y$. The area $A = xy=500000$, so $y=\frac{500000}{x}$. The total length of the fence $L = 3x + 2y$.

Step2: Substitute $y$ into $L$

Substitute $y=\frac{500000}{x}$ into $L$, we get $L(x)=3x + 2\times\frac{500000}{x}=3x+\frac{1000000}{x},x>0$.

Step3: Find the derivative

Differentiate $L(x)$ with respect to $x$. Using the power - rule, $L^\prime(x)=3-\frac{1000000}{x^{2}}$.

Step4: Set the derivative equal to zero

Set $L^\prime(x) = 0$, then $3-\frac{1000000}{x^{2}}=0$. Rearranging gives $3x^{2}=1000000$, so $x^{2}=\frac{1000000}{3}$, and $x=\frac{1000}{\sqrt{3}}$.

Step5: Find the second - derivative

Differentiate $L^\prime(x)$ to get the second - derivative $L^{\prime\prime}(x)=\frac{2000000}{x^{3}}$. When $x = \frac{1000}{\sqrt{3}}$, $L^{\prime\prime}(x)>0$, so $L(x)$ has a minimum at $x=\frac{1000}{\sqrt{3}}$.

Step6: Find the value of $y$

Substitute $x=\frac{1000}{\sqrt{3}}$ into $y=\frac{500000}{x}$, we get $y=\frac{500000}{\frac{1000}{\sqrt{3}}}=500\sqrt{3}$.

Step7: Calculate the minimum length of the fence

Substitute $x=\frac{1000}{\sqrt{3}}$ into $L(x)$: $L = 3\times\frac{1000}{\sqrt{3}}+2\times500\sqrt{3}=\frac{3000}{\sqrt{3}} + 1000\sqrt{3}=1000\sqrt{3}+1000\sqrt{3}=2000\sqrt{3}\approx3464.1$ feet.

Answer:

$2000\sqrt{3}\approx3464.1$ feet