The template below shows how to create a box with a lid from a sheet of cardboard that is 10 inches wide and 24 inches long. In the template, solid line segments indicate cuts, and dashed line segments indicate folds. The square flaps, each with a side length of x inches, are glued to the sides and overlap when the box is folded up. The box has 1. Before someone folds up the cardboard into a box, they look at the flat piece of cardboard and a bottom and four upright sides. The lid, which is attached to one of the upright sides, has three upright sides of its own. Assume that the three sides of the lid can be tucked wonder how much cardboard (square inches) is being used. What is the area of the flat cardboard inside the box when the lid is closed. A Box Company is looking at how to model this in order to find ways to sell their boxes and increase profits. in square inches? B i U Font Family - AA- A D F . E . E . 10 in. -24 in. - 2. Show the mathematics you used to find your answer. B i U Font Family - AAA D F . E . E .The cardboard can be folded into a box that looks like the one shown below. The company has determined that the height of the box will be x inches and they would like to determine the other dimensions of the box. Given that the height of the box is x inches, identify the width and length of the box in terms of x. Q 3. Width: Bit U Font Family - AA A +V Q 4. Length: B i U Font Family - AA A DE . E . E .The function V, = 3x - 39x* + 120x represents the volume of the box in cubic inches and can be modeled by the graph below. Use the graph to answer N changes saved questions the following questions. Q) 5. The graph that shows the volume of the box has been broken into four different colors: black, blue, green, and red. Complete the table below by naming the color that -100 corresponds with the domain it represents. 1. x 8 6. What is the domain for the function Vx=3x3-39x2+120x that represents the volume of the box? 7. Describe why you believe the domain you chose represents all of the reasonable values of x for the situation? B i U Font Family - AA. A D E . E . E . +V) 8. The Box Company wants to produce the box that has the greatest possible volume. What value of height (x) would produce a box with the maximum possible volume? ) 9. What are the dimensions of a box that has the x value you chose in the previous question? Width: ) 10. What are the dimensions of a box that has the x value you chose in the previous question? Length:11. What is the Volume of that box? Use a piece of paper to show the mathematics you used to find your answer. Take a screenshot of your work and attach it as an image in the space below. B i U Font Family - AA- A tvQ 12. Explain why the Box Company might want to know what the maximum volume would be for their boxes. B i U Font FamilyThe Box Company is wondering if they can use the same amount of cardboard to create a box with more volume when it is folded. The cardboard is 8 inches wide and 30 inches long. A picture of the cardboard is displayed below. 13. Do you agree or disagree with this claim? Agree Disagree 8 in. 30 in. 14. Write an explanation to justify your answer and include the mathematics you used. A worker at the Box Company claims that since this new cardboard layout has the same area as the original cardboard layout in square inches, the box that it makes when it is folded will also have the same volume (in cubic inches) as the original box. B i U Font Family - AAT A O F . E . E .Help the Box Company create a new box that has a larger volume without changing the area of the flat cardboard. The dimensions of the cardboard can change, but the area of the flat cardboard must be the same as the original. Create a new template to help the company find new dimensions (length and ) 15. On a piece of paper, draw the diagram and write the dimensions of the cardboard in the red width) for cardboard that would fold into a box that has a greater volume than the original box. boxes. Take a screenshot of your diagram and attach it as an image in the space below. B i U Font Family - AA- A D E . E . E . #V 16. Explain why your new dimensions will help the Box Company have a box with a greater volume than that of the original box. B i U Font Family - AA- A D E . E . E