You want to create a rectangular fenced-in area for your new pet chickens that needs to be 100 square feet. Additionally, you want the pen to be divided into 2 sections so you can separate your chickens as needed. You plan to use an opaque fencing material that costs $10 per linear foot on just the divider portion (so your chickens can't see the others). For the other sides, you will use a cheaper material that costs $2 per linear foot.

Determine the dimensions that you will need to use in order to minimize cost.

With a desired area of 100 square feet and two different types of fencing material, the optimal dimensions for the fenced area will be width of sqrt(50) feet and length of 2*sqrt(50) feet. This ensures the pricey opaque divider is used relatively less, thus minimizing costs.

Explanation:

To minimize the cost of the fencing while ensuring a total area of 100 square feet, one option would be to create a rectangular area with the length (l) being twice the value of the width (w). This is because the cost is smaller at the width sides and you would want to minimize the length of the expensive divider.

Given l = 2w and the area (lw) = 100, we can substitute l = 2w in the area equation becoming 2w*w = 100. Solving this equation, we find that: w = sqrt(50) and l = 2w = 2*sqrt(50).

So, the optimal dimensions to minimize the cost will be a rectangle of width sqrt(50) feet and length 2*sqrt(50) feet.

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