Final answer:
To find the maximum possible volume of the rectangular box, we need to consider the given information. The box has a square base with an edge of at least 1 inch long. It has no top, so it only has five sides. The total area of these five sides is 300 in². To maximize the volume, we need to find the maximum value for 'x' and 'h' that satisfies the equation: 4x² + 2xh = 300.
Step-by-step explanation:
To find the maximum possible volume of the rectangular box, we need to consider the given information.
The box has a square base with an edge of at least 1 inch long. It has no top, so it only has five sides.
The total area of these five sides is 300 in².
Let's assume the base is a square with an edge length of 'x' inches. Since the base has four sides, the total area of the base is 4x² in².
The other side of the box is a rectangle with a length of 'x' inches and a height of 'h' inches.
The total area of this side is 2xh in².
Now, we have the equation: 4x² + 2xh = 300.
To maximize the volume, we need to find the maximum value for 'x' and 'h' that satisfies this equation.
Given that the base has to be a square with an edge of at least 1 inch long, the smallest possible value for 'x' is 1 inch. Substituting 'x' with 1 in the equation, we have: 4(1)² + 2(1)h = 300.
Simplifying, we get: 4 + 2h = 300.
Solving for 'h', we find h = 148 inches.
The maximum possible volume occurs when 'x' = 1 inch and 'h' = 148 inches.
The volume of the box is V = x²h.
Substituting the values, we get V = 1²(148) = 148 cubic inches.