3. (2 points) A campground owner has 1400 meters of fencing. He wants to enclose a rectangular field bordering a river, with no fencing needed along the river. Find the dimensions of the field that will maximize the area.

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UpStudy AI · Accepted Answer

**Step 1. Define the variables**

Let:
- $$ x $$ be the length (in meters) of the side perpendicular to the river.
- $$ y $$ be the length (in meters) of the side parallel to the river (which is not fenced).

Since the field is bordered by the river along one side, fencing is required only for the other three sides:
$$
2x + y = 1400
$$

**Step 2. Express the area in terms of one variable**

The area $$ A $$ of the rectangle is:
$$
A = x \times y
$$
Solve the fencing constraint for $$ y $$:
$$
y = 1400 - 2x
$$
Then substitute $$ y $$ into the area equation:
$$
A = x(1400 - 2x) = 1400x - 2x^2
$$

**Step 3. Find the value of $$ x $$ that maximizes $$ A $$**

To maximize $$ A $$, differentiate with respect to $$ x $$ and set the derivative equal to zero:
$$
\frac{dA}{dx} = 1400 - 4x = 0
$$

Solve for $$ x $$:
$$
1400 - 4x = 0 \quad \Longrightarrow \quad 4x = 1400 \quad \Longrightarrow \quad x = 350
$$

**Step 4. Determine the corresponding value of $$ y $$**

Substitute $$ x = 350 $$ back into the equation for $$ y $$:
$$
y = 1400 - 2(350) = 1400 - 700 = 700
$$

**Final Answer**

The dimensions that maximize the area are:
$$
\textbf{Perpendicular to the river: } 350 \text{ meters} \quad \text{and} \quad \textbf{Parallel to the river: } 700 \text{ meters.}
$$
To maximize the area, the field should be 350 meters perpendicular to the river and 700 meters parallel to the river.

(2 points) A campground owner has 1400 meters of fencing. He wants to enclose a rectangular field bordering
a river, with no fencing needed along the river. Find the dimensions of the field that will maximize the area.

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