Final answer:
The formula for the maximum volume of a suitcase with a fixed height on an airplane is V = l × (57 - l - h) × h. To maximize the volume, the height should be set at 19 inches, which makes the length and width also equal to 19 inches.
Step-by-step explanation:
To find the maximum volume of a suitcase with a fixed height that can be carried on an airplane where the sum of the length, width, and height must not exceed 57 inches, we can employ the use of calculus and optimization techniques. Let's denote the height of the suitcase as h, which is fixed, and the width and length as w and l respectively. The constraint given is l + w + h = 57 inches.
The volume, V, of the suitcase can be represented as V = l × w × h. Since h is fixed, we can express w in terms of l and h using the constraint: w = 57 - l - h. Substituting this into the volume formula gives V = l × (57 - l - h) × h. This formula represents the volume as a function of length l, given a fixed height h.
To maximize the volume, we set the derivative of V with respect to l equal to zero and solve for l. Doing this gives l = (57 - h) / 2. Then, substituting l back into the constraint, we get w = (57 - h) / 2 as well. The height that allows the maximum volume is when l, w, and h are equal, which happens when h = 57 / 3 = 19 inches.