A farmer has 1200 m of fencing and wants to fence off a rectangular field that borders a straight river. He does not need to fence along the river. What are the dimensions of the field that has the…

Solution to the Fencing Problem

Problem Statement Recap

A farmer has 2000 ft of fencing and wants to enclose a rectangular field that borders a straight river. Since the river acts as one side of the rectangle, no fencing is required along that side. We need to find the dimensions of the rectangle that maximize the enclosed area.

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Step 1: Define Variables and Constraints

Let:

• $x$ = length of the side parallel to the river (ft).
• $y$ = length of the side perpendicular to the river (ft).

Since the river provides one boundary, fencing is only required for:

• The side parallel to the river: $x$ (no fence needed on the river side).
• The two sides perpendicular to the river: $2y$.

Total fencing used: $$x+2y=2000\text{(Constraint)}$$

Step 2: Express Area in Terms of One Variable

The area $A$ of the rectangle is: $$A=x\cdot y$$

From the constraint, solve for $x$: $$x=2000-2y$$

Substitute into the area formula: $$A(y)=(2000-2y)\cdot y=2000y-2y^2$$

Step 3: Find the Maximum Area Using Calculus

To find the maximum area, take the derivative of $A(y)$ with respect to $y$ and set it to zero: $$\frac{dA}{dy}=\frac{d}{dy}(2000y-2y^2)=2000-4y$$

Set the derivative equal to zero for critical points: $$2000-4y=0⟹4y=2000⟹y=500\text{ ft}$$

Step 4: Verify the Critical Point is a Maximum

Check the second derivative: $$\frac{d^{2}A}{d{y}^2}=-4<0$$ Since the second derivative is negative, $y=500$ ft corresponds to a maximum.

Step 5: Find the Corresponding $x$

Substitute $y=500$ back into the constraint: $$x=2000-2(500)=2000-1000=1000\text{ ft}$$

Step 6: Calculate the Maximum Area

$$A=x\cdot y=1000\cdot 500=500,000{\text{ ft}}^2$$

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Conclusion

The dimensions that maximize the area are:

• Length parallel to the river ($x$): 1000 ft
• Length perpendicular to the river ($y$): 500 ft

This configuration yields the largest possible area of 500,000 square feet with the given fencing constraint.