A parcel delivery service will deliver a package only if the length plus girth (distance around) does not exceed 24 inches.
(A) Find the dimensions of a rectangular box with square ends that satisfies the delivery service's restriction and has maximum volume. What is the maximum volume?
(B) Find the dimensions (radius and height) of a cylindrical container that meets the delivery service's restriction and has maximum volume. What is the maximum volume?
(A) The dimensions of the rectangular box are $\square$ in.
(Use a comma to separate answers as needed.)

Step 1 ：The problem is asking for the dimensions of a rectangular box with square ends that has the maximum volume under the condition that the length plus girth (distance around) does not exceed 24 inches. The girth of a rectangular box with square ends is 4 times the side of the square end. So, the condition can be written as \(L + 4W \leq 24\), where \(L\) is the length and \(W\) is the width of the square end. The volume of such a box is \(L*W^2\). We need to maximize this volume under the given condition.

Step 2 ：Let's denote the side of the square end as \(x\). Then the length of the box is \(24 - 4x\). The volume \(V\) as a function of \(x\) is \(V(x) = x^2 * (24 - 4x)\). We need to find the maximum of this function on the interval [0, 6].

Step 3 ：By taking the derivative of the volume function and setting it equal to zero, we find the critical points of the function. These are the values of \(x\) where the volume could potentially be maximized or minimized.

Step 4 ：The critical points are \(x = 0\) and \(x = 4\). We also need to check the volume at the endpoints of the interval, which are \(x = 0\) and \(x = 6\).

Step 5 ：The volumes at these points are \(V(0) = 0\), \(V(6) = 0\), \(V(4) = 128\). So, the maximum volume is achieved when \(x = 4\).

Step 6 ：The dimensions of the box are \(x, x, 24 - 4x = 4, 4, 8\) inches. The maximum volume is \(128\) cubic inches.

Step 7 ：Final Answer: The dimensions of the rectangular box are \(\boxed{4, 4, 8}\) in. The maximum volume is \(\boxed{128}\) cubic inches.