Chocolate Box Company is going to make open-topped boxes out of 7 × 11-inch rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way? (Round your answer to one decimal place.)

what are the answer choices?

the volume v is

v = x(7-2x)(11-2x) = 4x^3-36x+77x

so, find x where dv/dx = 0

no answer choices

To find the largest volume box that can be made from the given rectangles, we need to maximize the volume by choosing the appropriate size for the cut squares.

Let's assume that 'x' is the length of each side of the cut square.

If we cut 'x' squares from each corner of the rectangle, the resulting dimensions of the box will be:
Length: 11 - 2x
Width: 7 - 2x
Height: x

The volume of the box can be calculated by multiplying the three dimensions:
V = (11 - 2x)(7 - 2x)(x)

To find the maximum volume, we can differentiate the volume function with respect to 'x' and set it equal to zero:

dV/dx = 0

Let's calculate the derivative and solve for 'x':

dV/dx = (11 - 2x)(7 - 2x)' + (11 - 2x)'(7 - 2x)(x)' = 0

Expanding and simplifying the derivative equation gives us:

-4x^2 + 36x - 77 = 0

Now we can solve this quadratic equation for 'x'. Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = -4, b = 36, and c = -77, we can substitute these values into the formula and solve for 'x'.

so, do the math!

what is dv/dx?