Consider an open-top box constructed from an 8.5 × 11 inch piece of paper by cutting out squares of

equal size at the corners, then folding up the resulting flaps. Denote by x the side-length of each cut-out
square.
a) Draw a picture of this construction, and find a formula for the volume V of the box as a function
of x.
b) State the domain of V (x). (The domain is not all real numbers!)
c) Scaling appropriately, use your calculator to sketch the graph of V .

width = 8.5-2x

length = 11 - 2x
height = x

V = x(8.5-2x)(11-2x)
domain: 0 < x < 4.25

a) To construct the open-top box, we start with an 8.5 x 11 inch piece of paper. We cut out squares of equal size from each corner and fold up the resulting flaps to form the sides of the box.

Let's denote the side length of each cut-out square as x.

To visualize the construction, draw a rectangle with dimensions 8.5 x 11 inches.

11 inches
--------------
| |
| |
8.5| |
| |
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Next, from each corner of the rectangle, cut out a square with side length x. Fold up the flaps to form the sides of the box.

11 inches
--------------
| X X X |
| X X |
8.5| X X |
| X X X |
--------------

Now the sides of the box are formed by the flaps folded up. The length of the base of the box will be the original length of the rectangle minus the lengths of the two flaps on each side. Similarly, the width of the base of the box will be the original width of the rectangle minus the widths of the two flaps on each side.

The height of the box will be equal to the side length of each cut-out square.

Therefore, the dimensions of the base of the box will be (8.5 - 2x) inches by (11 - 2x) inches, and the height of the box will be x inches.

The volume V of the box can be calculated by multiplying the dimensions of the box:

V(x) = (8.5 - 2x)(11 - 2x)(x)

b) The domain of V(x) represents the possible values of x, which in this case is the side length of the cut-out square.

For the formula to make sense, the side length x must be positive and smaller than half of the length and width of the rectangle, to ensure that the resulting flaps can be folded up to create a box.

So, the domain of V(x) is 0 < x < min(8.5/2, 11/2).

c) To sketch the graph of V on a calculator, you need to scale the axes appropriately. Since the original dimensions are given in inches, the resulting volume may be too large to fit on the graph if we plot it directly.

One possible approach is to choose a suitable scale for the x-axis (side length) and the y-axis (volume). For example, you could set the x-axis scale to be 0.1 inch per unit and the y-axis scale to be 1 cubic inch per unit.

Using these scales, you can plot points for different values of x by evaluating the formula V(x), and then connecting the points to sketch the graph of V(x).

a) To construct the open-top box, start with an 8.5 × 11 inch piece of paper. Cut out a square from each corner of the paper with side length x. After cutting out the squares, you will have four flaps along the edges of the paper. Fold these flaps up to form the sides of the box. The dimensions of the box will be as follows:

- The length of the base of the box will be 11 - 2x inches, since you have removed two squares, each with side length x, from the 11-inch side.
- The width of the base of the box will be 8.5 - 2x inches, following the same reasoning as above.
- The height of the box will be x inches, as this is the length of each flap that was folded up.

So, the volume V of the box can be calculated by multiplying the length, width, and height of the box:

V(x) = (11 - 2x)(8.5 - 2x)(x)

b) To determine the domain of V(x), we need to consider the restrictions on the value of x. Since we are cutting squares out of the corners of the paper, the side length x cannot be greater than half of the length or width of the paper. Therefore, the domain of V(x) is limited by the condition:

0 ≤ x ≤ min(11/2, 8.5/2)

So, the domain of V(x) is 0 ≤ x ≤ 4.25.

c) To sketch the graph of V, we can scale the dimensions appropriately and use a calculator to evaluate the volume for several values of x within the domain. Choose a suitable range for x, such as 0 to 4.25, with increment steps of 0.25.

1. Substitute each value of x into the expression for V(x) and calculate the corresponding volume.
2. Plot the pairs of values (x, V) on a graph.
3. Connect the points with a smooth curve to obtain the sketch of the graph of V.

By following these steps, you can use a calculator or computer software to generate a visual representation of the graph of V(x).