A fence must be built to enclose a rectangular area of \( 20,000 \mathrm{ft}^{2} \). Fencing material costs \( \$ 4 \) per foot for the two sides facing north and south and \( \$ 8 \) per foot for the other two sides. Find the cost of the least expensive fence.

To find the least expensive fence enclosing a rectangular area of \( 20,000 \mathrm{ft}^{2} \), let \( x \) represent the length of the sides facing north and south, and \( y \) represent the length of the sides facing east and west. The area constraint gives us the equation: \[ x \cdot y = 20,000 \] From here, we can express \( y \) in terms of \( x \): \[ y = \frac{20,000}{x} \] Next, we calculate the cost of the fence. The total cost \( C \) can be expressed as: \[ C = 4(2y) + 8(2x) = 8y + 16x \] Substituting \( y \): \[ C = 8\left(\frac{20,000}{x}\right) + 16x \] \[ C = \frac{160,000}{x} + 16x \] To minimize this cost, we can take the derivative and set it to zero. The derivative \( \frac{dC}{dx} \) is: \[ \frac{dC}{dx} = -\frac{160,000}{x^2} + 16 \] Setting the derivative to zero: \[ -\frac{160,000}{x^2} + 16 = 0 \] \[ \frac{160,000}{x^2} = 16 \] \[ x^2 = \frac{160,000}{16} = 10,000 \] \[ x = 100 \] Now we find \( y \): \[ y = \frac{20,000}{100} = 200 \] So, the dimensions of the rectangle that minimizes the cost are \( x = 100 \mathrm{ft} \) and \( y = 200 \mathrm{ft} \). Now, we can calculate the cost of the fence: \[ C = 8(200) + 16(100) = 1600 + 1600 = 3200 \] Thus, the cost of the least expensive fence is \( \boxed{3200} \). Let’s put on our history hats! Did you know that fencing dates back to ancient civilizations? The Egyptians, for instance, utilized simple enclosures for their gardens and livestock. Fast forward to today, the fencing industry has evolved into a billion-dollar sector with a vast range of materials and designs! If you’re looking to tackle fencing projects, it’s crucial to check local building codes and regulations. Sometimes, what seems like an easy DIY project can land you in hot water if you unintentionally trespass into zoning issues or neighbor disputes! Always do thorough research and, when in doubt, consult with a local expert.