Interpreting a Model: Minimizing Surface Area. A company wants to design an open- topped box with with a square base that will have a volume of 2000 cubic inches. Suppose the company wants to design the box so that it uses the least amount of material. Let x represent the length of one of the sides on the base and h represents the heights of the box. A. Determine an equation for the surface area, A(r), of the box in terms of z. B. Consider the graph of y = A(x) (please use this graph to help make sure you have part A. correct). 4000 -3500 -3006 2500 2000 (5, 1625) 500 (25, 945 1000 -506 (15.874, 755.953) 12 14 16 18 20 22 24 26 28 30 Suppose you are an engineer that has developed the equation and graph modeling the surface area of the box. (i) The CEO puts a finger on the point (5, 1625) on the graph, and asks "what does that point mean?" In the context of the project, explain what that point means regarding the properties of the box. Includes units where appropriate and the dimensions of the box. (ii) The CEO puts a finger on the point (25, 945) on the graph, and asks "what does that point mean?" In the context of the project, explain what that point means regarding the properties of the box. Includes units where appropriate and the dimensions of the box. (iii) The CEO traces the graph as a approaches 0, and asks "what's going on here?" Explain how the box would change as a gets smaller and smaller, and be sure to justify why there is a vertical asymptote at = = 0. (iv) The CEO traces the graph as I approaches co, and asks "what's going on here?" Explain how the box would change as r gets larger and larger, and be sure to justify why the behavior of the graph is consistent with your description of how the box is changing. (v) Explain to the CEO why the point (15.874, 755.953) is important to the company? (vi) The engineering team realize they machine designed to build the box is incredibly precise and can cut the dimensions accurate to 5 decimal points. This would require you to improve the estimate of 15.874. Find the minimum point on the curve accurate to 5 decimal points by solving A'(x) = 0