Lindsey has 45ft of metal fencing to form 3 sides of a rectangular garden. a tall wooden fence already in place will serve as the fourth side. what lengths of fencing will produce the largest area?

To answer this item, we let L be the length and W be the width. The area of the rectangles is calculated through the equation,

           Area = L x W

From the given scenario, we can deduced that,
              L + 2W = 48 

Such that L = 48 - 2W.

Substituting this to the equation for area,
           Area = (48 - 2W)(W)

Simplifying,
            Area = 48W - 2W²

Differentiating and equating to zero,
             dArea = 48 - 4W = 0

The value of W from the equation is 12. Hence,
           L = 48 - 2W = 48 - 2(12) = 24

The dimensions are:
             Length = 24 ft
              Width = 12 ft