Questions: A farmer wants to construct a fence around an area of 15 f t2 in a rectangular field and then subdivide it in half with a fence parallel to one of the sides of the rectangle. What dimensions should the fenced area have in order to minimize the length of fencing used?

Solution Steps

To minimize the length of the fencing used, we need to express the total length of the fence in terms of the dimensions of the rectangle and then find the dimensions that minimize this expression. Let the length and width of the rectangle be \( l \) and \( w \), respectively. The area constraint gives us \( l \times w = 15 \). The total length of the fence includes the perimeter of the rectangle and the additional fence to subdivide it, which is \( 2l + 3w \). We will use the area constraint to express one variable in terms of the other and then find the minimum of the fencing function.

We need to minimize the total length of fencing used for a rectangular area of \( 15 \, \text{ft}^2 \). Let the dimensions of the rectangle be \( l \) (length) and \( w \) (width). The area constraint is given by: \[ l \cdot w = 15 \]

The total length of the fencing \( L \) includes the perimeter of the rectangle and an additional fence to subdivide it: \[ L = 2l + 3w \]

Using the area constraint \( w = \frac{15}{l} \), we can express the total length \( L \) in terms of \( l \): \[ L = 2l + 3\left(\frac{15}{l}\right) = 2l + \frac{45}{l} \]

By minimizing the function \( L \), we find the optimal dimensions. The optimization yields: \[ l \approx 4.743 \quad \text{and} \quad w \approx 3.162 \]

Final Answer