An open-top box is to be constructed from a 6 foot by 8 foot rectangular card- board by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out. Find the function f (9:) that represents the volume of the box in terms of x. This project is designed to experiment and hopefuly lead to a deeper un derstanding of the epsilon delta definition of limits. In particular, one goal is to help the student see that the definition is built around a matter of consider- able practical concern: Suppose I know that the output y (the volume of the box) of some factory, is related to the input m (in inches) by a functional relation 1,) = f(x) (in cubic inches). If we want our output y to be within some specified margin of error, which we will call epsilon, of a desired output value Yo, then how closely do we need to control the margin of errorr on the input which is what we call delta to achieve the desired accuracy in our output? That is, if I want three decimal places of accuracy on my output, how much accuracy do I need on my input? We will also reveal an important connection between slopes of tangent lines and epsilon-delta ratios in the project. Here are the questions: (1) What is the side length of the cardboard if the box volume is 10.0 cubic inches? (2) If the side length varies by 10.01 inches, what is the resulting range of values for the volume? (3) Next, consider the volume of the box drawn below. Let a: be the side length of the rectangular cradboard used to make the box. Find the volume f(z) of the box as a function of z, and set 5 = 0.0675 = 24. Then, for :1: = k with k = 1, 2, 3, . . . ,6, find the limit 1113in (37) and call them L1,L2,L3,L4,L5, L6. Then figure out what is the the smallest 6 of the form 6 = 2\"" such that II lc|