Questions: Enclosing the Largest Area The owner of the Rancho Los Feliz has 3480 yd of fencing to enclose a rectangular piece of grazing land along the straight portion of a river and then subdivide it by means of a fence running parallel to the sides. No fencing is required along the river. (See the figure below.) What are the dimensions of the largest area that can be enclosed?

Solution Steps

To maximize the area of a rectangular piece of land with a given amount of fencing, we can use calculus or algebraic methods. Since one side of the rectangle is along the river, we only need to consider three sides of the rectangle for the fencing. Let the length parallel to the river be \( x \) and the width perpendicular to the river be \( y \). The total fencing used is \( x + 2y = 3480 \). We need to express the area \( A = x \times y \) in terms of one variable and then find the maximum area using optimization techniques.

1. Express \( y \) in terms of \( x \) using the fencing constraint: \( y = \frac{3480 - x}{2} \).
2. Substitute \( y \) in the area formula: \( A = x \times \frac{3480 - x}{2} \).
3. Simplify the area function and find its derivative.
4. Find the critical points by setting the derivative to zero and solve for \( x \).
5. Determine the maximum area by evaluating the area function at the critical points.

Let \( x \) be the length of the side parallel to the river, and \( y \) be the width perpendicular to the river. The total amount of fencing available is given by the equation: \[ x + 2y = 3480 \]

From the fencing constraint, we can express \( y \) in terms of \( x \): \[ y = \frac{3480 - x}{2} \] The area \( A \) of the rectangle can then be expressed as: \[ A = x \cdot y = x \cdot \left(\frac{3480 - x}{2}\right) = \frac{3480x - x^2}{2} \]

To find the maximum area, we take the derivative of \( A \) with respect to \( x \): \[ A' = \frac{d}{dx}\left(\frac{3480x - x^2}{2}\right) = 1740 - x \] Setting the derivative equal to zero to find critical points: \[ 1740 - x = 0 \implies x = 1740 \]

Substituting \( x = 1740 \) back into the equation for \( y \): \[ y = \frac{3480 - 1740}{2} = 870 \] The maximum area can be calculated as: \[ A = x \cdot y = 1740 \cdot 870 = 1513800 \]

Final Answer