an 18 inch by 20 inch sheet of cardboard is cut and folded to make a box for the great pecan company.

a. write a polynomial funtion to model the volume of the box.

how can i graph this?

c. the company wants the box to have a volume of 224 cubic inches. write an equation to model this situation.
d. find a positibe integer for x.

I will asssume that you are going to cut equal squares from each of the four corners to do this.

Let each side of a square be x inches.

then the length of the box will be 20-2x
its width will be 18-2x in
and its height will be x in

since Vol = length x width x height

Vol = x(20-2x)(18-2x)

make a table of values, noting there are zeros at x=0, x=10 and x=9

b) x(20-2x)(18-2x) = 224

c) expanding this and simplifying I got

x^3 - 19x^2 + 90x - 56=0

I ran this through the following cubic equation solver

http://www.1728.com/cubic.htm

and got x=6.529, the other two answers are extraneous.

(check my work, I tend to get careless at this time of night)

a. The volume of the box can be modeled using a polynomial function by considering the dimensions of the sheet of cardboard being cut and folded. Let's assume that the dimensions of the box are x, y, and z.

Since the cardboard is cut and folded, the length and width of the box (x and y) will be the dimensions of the original cardboard sheet, while the height (z) will be the height of the cut-out flaps.

Therefore, the polynomial function to model the volume of the box can be written as:

V(x,y,z) = (x - 2z) * (y - 2z) * z
where V represents the volume.

b. To graph this polynomial function, you will need to choose appropriate values for x and y, and then calculate the corresponding value of z using the given dimensions of the cardboard sheet. This will give you the volume at different points.

For example, if the dimensions of the cardboard sheet are 18 inches by 20 inches, you can choose different values for x and y, while keeping the z value within reasonable limits. Plot the values on a graph with the volume (V) on the y-axis and x (or y) on the x-axis to create a 3D plot.

c. If the company wants the box to have a volume of 224 cubic inches, you can use the equation from part a and set it equal to the desired volume:

V(x,y,z) = (x - 2z) * (y - 2z) * z = 224

d. To find a positive integer value for x, you will need to substitute values for y and z into the equation from part c and solve for x. However, since we do not have specific values for y and z, it is not possible to find a specific positive integer solution for x.

a. To find the polynomial function that models the volume of the box, we need to understand the dimensions of the box formed from the cardboard sheet.

Let's assume that the sheet of cardboard is cut to form a rectangular box by removing squares of side length 'x' from each corner, and then folding the remaining flaps upward. The length of the box will be (18 - 2x) inches, and the width will be (20 - 2x) inches. Lastly, the height of the box will be 'x' inches.

Therefore, the volume of the box can be calculated as the product of its length, width, and height:
V = (18 - 2x)(20 - 2x)(x)

b. To graph this polynomial function, you can use a graphing calculator or software like Desmos. Follow these steps:
1. Open a graphing calculator or go to the Desmos website.
2. Enter the polynomial function V = (18 - 2x)(20 - 2x)(x) in the equation section.
3. Set the x-axis and y-axis ranges to appropriate values. In this case, you can start with x-values from 0 to 10, and y-values from 0 to 500 (or adjust based on your preference).
4. Click on the "Graph" button to plot the function.
5. You should see a graph representing the polynomial function. The x-intercepts of the graph will indicate possible positive integer solutions for 'x', as the dimensions of the box cannot be negative or non-integer.

c. The equation to model the situation where the company wants the box to have a volume of 224 cubic inches can be derived by substituting the desired volume into the previously mentioned polynomial function:

224 = (18 - 2x)(20 - 2x)(x)

d. To find a positive integer solution for 'x', you can apply different methods:
1. Analyzing the graph: Look for the x-intercepts on the graph, i.e., the points where the polynomial function crosses the x-axis. These intercepts will represent possible positive integer solutions for 'x'. You can find the corresponding 'x' values on the graph.
2. Numerical Methods: You can use numerical methods, such as trial and error or iterative methods, to find a positive integer solution for 'x'. By substituting different integers for 'x' into the equation, you can determine if it satisfies the equation (i.e., 224 = (18 - 2x)(20 - 2x)(x)). Keep trying different positive integers until you find a solution.

These methods should help you find a positive integer value for 'x' that satisfies the equation.