Here's the problem I was asked: You are visiting your parents on a farm. They have asked you to design a small rectangular-shaped vegetable garden along an existing wall in their backyard. They wish to surround it with a small fence to protect their plants from animals. You have 15 yard of fencing to enclose the garden. The fencing is sold in .6 yard panels. In order to grow as many vegetables as possible, your task is to design a fence to enclose the maximum possible area. Partial panels of fencing may not be used.

How many panels of fencing should you use along the width of the garden? and How many panels would you use along the length of the garden?

I am also to have an equation showing the maximum and minimum.

If each panel is 0.6 yards wide and you have fencing with 15 yards of width, you have 15/0.6 or 25 panels. 25 is an odd number; you need an even number of panels for a rectangle since you can't cut them. That means you use 24 panels and thrown one away. The best option would be 6 panels to a side, which would mean a square with side width 6x0.6 = 3.6 yards. The total area woould be 3.6^2 = 12.96 square yards. If you go with 7 x 5 panels, that would be 4.2 x 3.0 yards or 12.60 sq yd, which is not as good.

You did not need to use algebra to do this problem.

drwls, I don't think you really grasped the concept of answering this question.

To find the maximum possible area of the rectangular vegetable garden, we need to determine the dimensions (width and length) that will maximize the area within the given 15 yards of fencing.

Let's assume that the width of the garden is represented by "w" and the length by "l". Since the garden is rectangular and has a fence along all sides, we can calculate the amount of fencing used by adding up the lengths of all four sides of the garden:

2w + 2l = 15

Now let's solve this equation to find the relationship between w and l:

2w + 2l = 15
2l = 15 - 2w
l = (15 - 2w) / 2
l = 7.5 - w

Now we can express the area of the garden as a function of one variable, w:

Area = length × width
Area = (7.5 - w) × w
Area = 7.5w - w^2

To find the maximum area, we need to find the vertex of this quadratic equation. The vertex occurs at the value of w that maximizes the area.

The x-coordinate of the vertex can be found using the formula:
w = -b / (2a)

In our case, a is -1 since the "w^2" term has a coefficient of -1, and b is 7.5 since the "w" term has a coefficient of 7.5. Plugging these values into the formula, we get:

w = -7.5 / (2 * -1)
w = 7.5 / 2
w = 3.75

Since the width cannot be a fraction of a yard, we need to round this value to the nearest whole number, which is 4. Therefore, you should use 4 panels of fencing along the width of the garden.

To find the length, we can substitute the value of w back into the equation we derived earlier:
l = 7.5 - w
l = 7.5 - 3.75
l = 3.75

Since the length also cannot be a fraction, we round this value to the nearest whole number, which is also 4. Thus, you should use 4 panels of fencing along the length of the garden.

Therefore, the maximum area of the rectangular vegetable garden can be achieved by using 4 panels of fencing along both the width and length of the garden.