Final answer:
To find the dimensions of the rectangular box with the largest volume, we need to maximize the volume function while satisfying the constraint that the sum of the length, width, and height does not exceed 96. The largest volume is achieved when the dimensions of the box are 96 in for length, width, and height.
Step-by-step explanation:
To find the dimensions of the rectangular box with the largest volume, we need to maximize the volume function while satisfying the constraint that the sum of the length, width, and height does not exceed 96. Let's denote the length, width, and height of the box as x, y, and z respectively.
The volume of a rectangular box is given by the formula V = xyz. Our constraint is x + y + z ≤ 96. To find the dimensions that maximize the volume, we can use the method of optimization by finding critical points.
Taking the derivative of the volume function with respect to each variable, we get:
dV/dx = yz
dV/dy = xz
dV/dz = xy
To find the critical points, we need to solve the system of equations obtained by setting the partial derivatives equal to zero:
yz = 0
xz = 0
xy = 0
Solving these equations, we find that the critical points are (0, 0, 0), (0, 0, 96), (0, 96, 0), and (96, 0, 0).
Next, we need to check the volume at these critical points and at the endpoints of the constraint:
At (0, 0, 0): V = 0
At (0, 0, 96): V = 0
At (0, 96, 0): V = 0
At (96, 0, 0): V = 0
At (96, 0, 96): V = 96*0*96 = 0
At (96, 96, 0): V = 96*96*0 = 0
At (0, 96, 96): V = 0
At (96, 96, 96): V = 96*96*96 = 884,736
Hence, the largest volume is achieved when the dimensions of the box are 96 in for length, width, and height.
Therefore, the correct option is: 96 in, 96 in, 96 in.