By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 16 in. long and 6 in. wide, find the dimensions of the box that will yield the maximum volume. (Round your answers to two decimal places.)

(smallest value)

(largest value)

To find the dimensions of the box that will yield the maximum volume, we can express the volume of the box in terms of its dimensions and then maximize the volume function.

Let's denote the length of the square being cut out from each corner as x. Since identical squares are being cut out from each corner, the dimensions of the resulting box will be:

Length: (16 - 2x) in
Width: (6 - 2x) in
Height: x in

The volume of the box can be found by multiplying these dimensions:

Volume = (16 - 2x) * (6 - 2x) * x

Now, we can find the maximum volume by maximizing the volume function.

To find the maximum value, we can take the derivative of the volume function with respect to x and set it equal to zero. Then solve for x.

Let's differentiate the volume function:

dV/dx = [(16 - 2x) * (6 - 2x) * x]'
= [(96 - 32x - 12x + 4x^2) * x]'
= (4x^3 - 92x^2 + 576x)'
= 12x^2 - 184x + 576

Setting this derivative equal to zero:

12x^2 - 184x + 576 = 0

Now, we solve this quadratic equation.

Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

a = 12, b = -184, c = 576

x = (-(-184) ± √((-184)^2 - 4(12)(576))) / (2(12))
x = (184 ± √(33856 - 27648)) / 24
x = (184 ± √6208) / 24

Now let's calculate the values of x:

x ≈ 2.091 or x ≈ 12.242

Since the dimensions of the box cannot exceed the dimensions of the original cardboard, the value of x must be less than or equal to 3.

Therefore, the valid value of x for this problem is approximately x ≈ 2.091.

Now we can find the dimensions of the box that will yield the maximum volume:

Length = 16 - 2(2.091) ≈ 11.818 inches
Width = 6 - 2(2.091) ≈ 1.818 inches
Height = 2.091 inches

So, the dimensions of the box that will yield the maximum volume are approximately:
Length: 11.818 inches
Width: 1.818 inches
Height: 2.091 inches