You have 4L feet of fence to make a rectangular vegetable garden alongside the wall of your house, where L is a positive constant. The wall of the house bounds one side of the vegetable garden. What is the largest possible area for the vegetable garden?

Perimater=4L=2width+1length

or length=4L-2w

area=width(4L-2W)

darea/dw=(4L-2w) + w(-2)=0
width=L
length=2L

check my thinking

looks correct. The fence is divided equally among lengths and widths.

yes 2L^2 is correct

Well, with the limited information you've given me, I'd have to say the largest possible area for your rectangular vegetable garden is when each side of the rectangle is 2L. This way, you'd have a perimeter of 4L, which matches your available fence. However, I have to ask, are you sure you want a rectangular vegetable garden? I hear square vegetables are all the rage these days!

To find the largest possible area for the vegetable garden, we need to determine the dimensions of the rectangle that maximize the area. Let's break down the problem step by step:

1. Let's assume the length of the vegetable garden is L and the width is W.
2. The length of the garden is equal to the length of the wall of the house, which is L.
3. The remaining fencing is used to create the two other sides of the garden, so the overall fence length is: 2W + L.
4. Since the fence length is 4L, we can write the equation: 2W + L = 4L.
5. Solving the equation for W, we have: 2W = 3L, or W = (3L / 2).
6. Now we have the length L and the width W in terms of L.
7. The area of the rectangle is given by multiplying the length and width: A = L * W.
8. Substituting the values of L and W, we get: A = L * (3L / 2) = 3L^2 / 2.

So, the largest possible area for the vegetable garden is given by the equation A = 3L^2 / 2.