Answer:
Explanation:
To find the dimensions of the box with the greatest volume, we need to maximize the volume function, which is given by V(x) = x(12-2x)^2, where x is the length of the side of the square that is cut from each corner.
To do this, we take the derivative of V(x) with respect to x and set it equal to zero to find the critical points:
V'(x) = 12x(12-2x) - 4x(12-2x)^2 = 0
Simplifying this expression, we get:
12x(12-2x)(1-2x) = 0
Therefore, the critical points are x = 0, x = 6, and x = 3. We need to determine which of these values maximizes the volume function.
To do this, we evaluate the volume function at each of these critical points and at the endpoints of the interval [0,6]:
V(0) = 0
V(6) = 0
V(3) = 108
Therefore, the dimensions of the box with the greatest volume that can be made from a 12 cm x 12 cm metal square are 6 cm x 6 cm x 12 - 2(6) = 6 cm x 6 cm x 12 - 12 = 6 cm x 6 cm x 0 cm. In other words, the box has no height because all four sides have been folded up to form the top of the box.