a farmer wants to fence an area of 750 000 m² in a rectangular field and divide it in half with a fence…

Explanation:

Step1: Define variables

Let the length of the rectangle be $x$ and the width be $y$. The area $A = xy=750000$, so $y=\frac{750000}{x}$. The total length of the fence $L = 3x + 2y$ (assuming the dividing - fence is parallel to the side of length $x$).

Step2: Substitute $y$ into the fence - length formula

Substitute $y=\frac{750000}{x}$ into $L$: $L(x)=3x + 2\times\frac{750000}{x}=3x+\frac{1500000}{x},x>0$.

Step3: Find the derivative of $L(x)$

Using the power rule, if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $L(x)=3x+\frac{1500000}{x}=3x + 1500000x^{-1}$, $L^\prime(x)=3-1500000x^{-2}=3-\frac{1500000}{x^{2}}$.

Step4: Set the derivative equal to zero and solve for $x$

Set $L^\prime(x)=0$, then $3-\frac{1500000}{x^{2}} = 0$. Add $\frac{1500000}{x^{2}}$ to both sides: $3=\frac{1500000}{x^{2}}$. Cross - multiply to get $3x^{2}=1500000$, then $x^{2}=500000$, and $x = 500\sqrt{2}\text{ m}$.

Step5: Find the value of $y$

Substitute $x = 500\sqrt{2}$ into $y=\frac{750000}{x}$. So $y=\frac{750000}{500\sqrt{2}}=\frac{1500}{\sqrt{2}} = 750\sqrt{2}\text{ m}$.

Answer:

The dimensions of the rectangle should be $500\sqrt{2}\text{ m}$ and $750\sqrt{2}\text{ m}$, with the dividing fence parallel to the side of length $500\sqrt{2}\text{ m}$.