Questions: You have 120 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area? a. 900 sq ft b. 140 sq ft c. 30 sq ft

Solution Steps

To maximize the area of a rectangle with a fixed perimeter, we should use the formula for the perimeter of a rectangle, which is \(2 \times (\text{length} + \text{width}) = 120\). We can express the width in terms of the length and substitute it into the area formula \( \text{Area} = \text{length} \times \text{width} \). Then, we find the length that maximizes the area by taking the derivative and setting it to zero.

We are given a fixed perimeter of \( P = 120 \) feet for a rectangular region. We need to find the dimensions of the rectangle that maximize the enclosed area \( A \).

Using the perimeter formula for a rectangle, we have: \[ P = 2(\text{length} + \text{width}) \implies 60 = \text{length} + \text{width} \] Thus, we can express the width as: \[ \text{width} = 60 - \text{length} \]

The area \( A \) of the rectangle can be expressed as: \[ A = \text{length} \times \text{width} = \text{length} \times (60 - \text{length}) = 60\text{length} - \text{length}^2 \]

To maximize the area, we take the derivative of \( A \) with respect to \( \text{length} \) and set it to zero: \[ \frac{dA}{d\text{length}} = 60 - 2\text{length} = 0 \] Solving for \( \text{length} \) gives: \[ \text{length} = 30 \]

Substituting \( \text{length} = 30 \) back into the width expression: \[ \text{width} = 60 - 30 = 30 \] Now, we can calculate the maximum area: \[ A = 30 \times 30 = 900 \text{ sq ft} \]

Final Answer