Question

a wire that is centimeters long is shown below. the wire is cut into two pieces, and each piece is bent and formed into the shape of a square. suppose that the side length (in centimeters) of one square is , as shown below. (a) find a function that gives the total area enclosed by the two squares (in square centimeters) in terms of . (b) find the side length that minimizes the total area of the two squares. (c) what is the minimum area enclosed by the two squares?

Answer 1

The total area enclosed by the two squares can be found by adding the areas of the individual squares. To minimize the total area, the side length of each square should be 0 cm. The minimum area enclosed by the squares would be 0 square centimeters.

Step-by-step explanation:

(a) To find the total area enclosed by the two squares, we need to find the area of each square and add them together. Each square has a side length of . Therefore, the area of each square is ². So the total area is ² + ², which simplifies to².

(b) To minimize the total area, we need to find the value of that makes the area function as small as possible. Since the area function is quadratic, its minimum value occurs at the vertex of the parabola. The vertex occurs at = -b/2a. In this case, a = 1 and b = 2, so = - 2/2(1) = -1. Therefore, the side length that minimizes the total area is -1 cm. However, in the context of the problem, a negative length doesn't make sense. So the minimum side length would be 0 cm.