Final answer:
To find the dimensions of the box that give the largest possible volume, we can solve an optimization problem by maximizing the volume function subject to a constraint equation. The dimensions of the box that maximize the volume are 20 cm, 20 cm, and 20 cm. The maximum volume of the box is 8,000 cm^3.
Step-by-step explanation:
To find the dimensions of the box that give the largest possible volume, we can use the concept of optimization. Let's define the dimensions of the box as x, x, and h, where x is the length of the side of the square base and h is the height. The volume of the box can be expressed as V = x^2h.
Since we have a constraint on the total surface area of the box, we can write an equation in terms of x and h. The surface area of the box is given as 6400 cm^2, which can be expressed as:
6400 = x^2 + 4xh.
To find the dimensions that maximize the volume, we need to maximize the volume function V = x^2h, subject to the constraint equation 6400 = x^2 + 4xh.
To solve this problem, we can use the method of Lagrange multipliers. However, for simplicity, we can solve the constraint equation for h in terms of x:
h = (6400 - x^2) / (4x).
Substituting this into the volume function, we get:
V = x^2 * (6400 - x^2) / (4x).
Simplifying this expression, we get:
V = (1600x - x^3) / 4.
To find the maximum value of V, we can take the derivative of V with respect to x, set it equal to zero, and solve for x:
dV/dx = 1600 - 3x^2 = 0.
Solving this equation, we get x = 20. Substituting this value back into the constraint equation, we find h = 20. Therefore, the dimensions of the box that give the largest possible volume are 20 cm, 20 cm, and 20 cm.
The maximum volume can be calculated by substituting the values of x and h into the volume function V = x^2 * h:
V = (20)^2 * 20 = 8,000 cm^3.