If 1200 square centimeters of material is available to make a box with a square base and open top, find the largest possible volume of the box.
To find the largest possible volume of the box, we need to determine the dimensions of the box that will maximize its volume. Let's break down the problem.
Let's assume the side length of the square base is 's' centimeters. Since the box has an open top, its height will also be 's' centimeters.
The surface area of the box consists of the area of the square base plus the sum of the areas of the four sides. The side lengths of the four sides are all 's', and the area of each side is 's × s = s²'. Since there are four sides, the total surface area is 4s².
According to the problem, the total surface area available is 1200 square centimeters. Therefore, we have the equation:
4s² = 1200
To find the value of 's', we divide both sides of the equation by 4:
s² = 1200 / 4
s² = 300
Taking the square root of both sides, we get:
s = √300
Now that we have the value of 's', we can calculate the maximum volume of the box. The volume of a rectangular box is given by the formula:
Volume = base area × height
Since the base is a square, the formula can be simplified to:
Volume = s² × s
Substituting the value of 's' we found earlier:
Volume = (√300)² × √300
Volume = 300 × √300
Volume ≈ 17320.51 cubic centimeters
Therefore, the largest possible volume of the box is approximately 17320.51 cubic centimeters.