I am in the process of building a storage building Originall

?I am in the process of building a storage building. Originally my plan was to build 2 large units (see picture below), but after doing some market research I determined it would be better to build 14 smaller equal size units. I will be using the same material for both the interior and exterior walls. I have a total of 420 lineal feet of wall material. Let X be the width of each storage unit, and Y be the depth. Since each unit will be exactly the same, the front side of the storage building will be 14X in length. To maximize

the size of each storage unit, what would X have to be?

Solution

Length of the sheet = 420 feet

14x + 14y = 420

x + y = 30

Area = 14x * 14y (which needs to be maximized) = 196 xy,

substituting the constraint

Area = 196x(30-x)

dArea/dx = 196 * 30 - 392x

equation dArea/dx = 0, we get x=15

y = 15

Area = 196 * 15 * 15 = 44100 ft^2