A box designer has been charged with the task of determining the surface area of various open boxes (no lid) that can be constructed by cutting four equal-sized surface corners from an 8-inch by 11.5 inch sheet of cardboard and folding up the sides.
1. Determine a function that relates the total surface area, s, (measured in square inches) of the open box to the size of the square cutout x (measured in inches).
2. What is the domain and range of the function s?
3. What is the surface area when a 1" x 1" square is cut out?
4. What size square cutout will result in a surface area of 20 in?
5. What is the surface area of the box when the volume is maximized? (Calculator)

1) , 2) The domain of S is . The range of S is , 3) , 4) , 5)

Step-by-step explanation:

1) The function of the box is:

2) The maximum cutout is:

The domain of S is . The range of S is

3) The surface area when a 1'' x 1'' square is cut out is:

4) The size is found by solving the following second-order polynomial:

5) The equation of the box volume is:

The first derivative of the function is:

The critical points are determined by equalizing the derivative to zero:

The second derivative is found afterwards:

After evaluating each critical point, it follows that is an absolute minimum and is an absolute maximum. Hence, the value of the cutoff so that volume is maximized is:

The surface area of the box is:

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