Final answer:
To find the largest volume of the box, we need to maximize the dimensions of the box while maintaining the constraint of 36 inches for the sides of the cardboard. We can do this by finding the critical points of the volume equation and evaluating the volume at those points. The largest volume of the box is 1458 cubic inches.
Step-by-step explanation:
To find the largest volume of the box, we need to maximize the dimensions of the box while maintaining the constraint of 36 inches for the sides of the cardboard.
Since we are cutting out squares from each corner, the length and width of the box will be 36 inches minus twice the length of each square cut out.
Let's say the length of each square cut out is x inches. This means the length and width of the box will be (36 - 2x) inches. The height of the box will be x inches, since we are bending up the sides.
So, the volume of the box is given by V = (36 - 2x)(36 - 2x)x.
To find the largest volume, we need to find the maximum value of V. We can do this by finding the critical points of V and evaluating the volume at those points.
First, let's find the derivative of V with respect to x: dV/dx = 4x^2 - 144x + 1296.
Solving dV/dx = 0, we find that x = 9 inches.
To confirm that this is a maximum, we can analyze the concavity of V.
Taking the second derivative of V, we have d^2V/dx^2 = 8x - 144.
Evaluating this at x = 9, we find that d^2V/dx^2 = -16, which is negative.
Therefore, x = 9 inches corresponds to a maximum volume.
Substituting x = 9 into the volume equation, we find the largest volume:
V = (36 - 2(9))(36 - 2(9))(9)
= 1458 cubic inches.