Final answer:
To find the dimensions of the box that give the largest possible volume, we need to maximize the volume function. The dimensions of the box that give the largest possible volume are approximately 110.89 cm for the side of the square base and 73.93 cm for the height. The maximum value of the volume is approximately 974965.18 cm^3.
Step-by-step explanation:
To find the dimensions of the box that give the largest possible volume, we need to maximize the volume function. Let's denote the length of the side of the square base as 'x' and the height of the box as 'h'.
The surface area of the box is given by the sum of the area of the square base and the area of the four sides. Based on the given information, the equation for the surface area is:
SA = x^2 + 4xh = 6400 cm^2
To find the volume of the box, we can use the equation:
V = x^2 * h
Now, we need to express the volume equation in terms of a single variable to apply optimization techniques. We can solve the surface area equation for 'h' and then substitute it into the volume equation.
First, solve the surface area equation for 'h':
h = (6400 - x^2) / (4x)
Substitute the expression for 'h' into the volume equation:
V = x^2 * ((6400 - x^2) / (4x))
This can be simplified to:
V = (x^3 * (1600 - x^2)) / 4
To find the maximum volume, we need to find the critical points of the volume function. We can do this by taking the derivative of the volume equation and setting it equal to zero.
Take the derivative of the volume equation with respect to 'x':
dV/dx = (3x^2 * (1600 - x^2) - x^3 * 2x) / 4
Set the derivative equal to zero and solve for 'x':
0 = (3x^2 * (1600 - x^2) - x^3 * 2x) / 4
Simplifying the equation gives:
0 = 4800x^2 - 3x^4
Factor out 'x^2' to solve for 'x':
0 = x^2 * (4800 - 3x^2)
From this equation, we can see that 'x' can equal 0 or 'x' can be a solution to the equation '3x^2 - 4800 = 0'.
Solving the quadratic equation gives two values for 'x':
x = sqrt(4800/3) and x = -sqrt(4800/3)
Since length cannot be negative, we discard the negative solution and use 'x = sqrt(4800/3)' as the length of the side of the square base.
To find the corresponding height, we substitute this value of 'x' into the surface area equation and solve for 'h':
h = (6400 - (4800/3)) / (4 * sqrt(4800/3))
Calculating the values gives:
x = sqrt(4800/3) ≈ 110.89 cm
h ≈ 73.93 cm
Therefore, the dimensions of the box that give the largest possible volume are approximately 110.89 cm for the side of the square base and 73.93 cm for the height.
The maximum value of the volume can be found by substituting these values into the volume equation:
V = (110.89^3 * (1600 - (110.89^2))) / 4 ≈ 974965.18 cm^3