Question

You have 7 L 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.

Answer 1

The question asks about optimizing the area of a rectangular garden with a fixed length of fencing by calculating the dimensions. One side of the rectangle is along the house wall, and the calculations involve maximizing a quadratic equation representing the area.

Step-by-step explanation:

The question involves using a given length of fencing to create a rectangular vegetable garden along the side of a house. To maximize the area of the garden with a fixed amount of fence, one would typically create a rectangle where the house wall acts as one of the longer sides of the rectangle. There are various methods for determining the dimensions of the garden, but a common approach is to let one dimension be x and the other will then be (7L - 2x)/2, since the total amount of fence used on the three sides away from the house must equal 7L. Therefore, the total area A of the garden would be x multiplied by (7L - 2x)/2. This forms a quadratic equation when setting the area in terms of x, and one can use calculus or algebraic techniques (such as completing the square) to find the value of x that maximizes A.