If 2100 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Volume = cubic centimeters.
Assuming x = side of box.
2100 = 5x^2
Solve for x.
Then V = x^3
To find the largest possible volume of the box with the given constraints, we can use optimization techniques. Let's go step by step:
1. Define the variables: Let's define the side length of the square base as "x" (in centimeters). The height of the box will also be "x" since the box has an open top.
2. Determine the equation to optimize: We need to maximize the volume of the box. The volume of a rectangular prism is given by V = length × width × height. In our case, the length and width are both "x" since the base is a square. So, the volume equation becomes V = x × x × x = x^3.
3. Determine the constraint: We are given that 2100 square centimeters of material is available. To form the box, we need 5 sides (4 sides of the base and the height) since the top is open. The total amount of material used will be the surface area of these 5 sides, which is 5 times the area of the base (x^2). So, we have the constraint equation: 5x^2 = 2100.
4. Solve the constraint equation for "x": Rearranging the constraint equation, we have x^2 = 2100 / 5 = 420. Taking the square root gives x ≈ √420 ≈ 20.49 cm (rounded to two decimal places).
5. Calculate the largest possible volume: Now that we have the value of "x," we can substitute it back into the volume equation to find the largest possible volume. V = x^3 ≈ (20.49)^3 ≈ 8668.81 cubic centimeters (rounded to two decimal places).
So, the largest possible volume of the box is approximately 8668.81 cubic centimeters.