A farmer plans to enclose a rectangular region using part of his barn for one side and fencing for remaining sides. The area of the region is maximized when the width is 30 feet.
a.) write an equation that could model the area as a function of the width. Let P equal the amount of available fencing material
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To find an equation that models the area as a function of the width, we first need to understand the given information. The farmer plans to enclose a rectangular region using part of his barn for one side and fencing for the remaining sides.
Let's assume the length of the rectangle is L, and the width is W. The area of a rectangle is obtained by multiplying the length by the width. Therefore, the area A can be given by the equation: A = L * W.
However, we are given that the farmer will use part of his barn for one side, which means the length of the rectangle will be less than the total available fencing material P. Since the farmer is using fencing for the remaining sides, the perimeter of the rectangle (2L + W) should be equal to the amount of available fencing material P.
Based on this information, we can write the equation: P = 2L + W.
Now, we need to express the length L in terms of the width W so that we can substitute it back into the area equation. We are also given that the width W is 30 feet, which means L = P - 2W (since 2L + W = P).
Substituting L = P - 2W into the area equation, we have: A = (P - 2W) * W.
Therefore, the equation that models the area A as a function of the width W, with P representing the amount of available fencing material, is: A = (P - 2W) * W.