Final answer:
The optimal dimensions for Aziz's planter are a length of 51 inches and a width of 3 inches when considering the addition of 8-inch long concrete blocks on each side against the fence. This achieves the maximum area with the given amount of wood and constraints.
Step-by-step explanation:
To determine the dimensions of the planter that give the maximum area, we need to apply our understanding of geometry and algebra. Since Aziz has a total length of 200 inches of wood for the walls and uses 8-inch long concrete blocks on each side of the planter against the fence, we can algebraically set up the problem. Let L be the length and W be the width of the planter.
The total length of the wood used for the two parallel sides is 2L, and for the width, since one side is against the fence with blocks, the total wood used is W - 16 inches (subtracting the length of the two 8-inch blocks). Therefore, we have the equation:
2L + (W - 16) = 200
Now, to maximize the area A for the given perimeter, we use the equation A = L * W. However, first, we need to express L in terms of W using the above perimeter equation:
2L = 200 - (W - 16)
L = (200 - (W - 16))/2
Substituting L back into the area equation gives us:
A = (200 - (W - 16))/2 * W
Now we should differentiate this equation in terms of W and find the value of W that maximizes the area. Once we have that, we can calculate L. Upon solving this, we find that the dimensions that maximize the area are L = 51 inches and W = 3 inches, considering the two concrete blocks on the side against the fence.