Section+1.7

A square of side x inches is cut out of each corner of an 8 inch by 15 inch piece of board and the sides are folded up to form an open-top box. (A) Write the volume of the box as a function of x.  The volume of the box is v=x(8-2x)(15-2x). This is the formula because the length of the one side is 8 inches with 2 cuts in it that are x inches long each. That means the one side’s length is 8 inches - 2x. The other side’s length is 15 inches – 2x because there were also 2 sqares with the length of x taken out. When folded up. The length x of the square that was cut out now becomes the height. The formula for volume is Height x Length x Width and in this case that is x(8-2x)(15-2x). (B) Find the domain of V as a function of x.  The domain is [0,4]. Any values below 0 or greater than 4 will make the volume negative, which mathematically is not possible. The values 0 and 4 will make the volume 0 which is mathematically possible and any of the values in between will make the volume positive. (C) Graph V as a function of x over the domain found in part (B) and use the maximum finder on your graph to determine the maximum volume such a box can hold. Plug the equation x(8-2x)(15-2x) into your graphing calculator for y1 and press enter. Then change you window settings so that the graph will display correctly. Change the x min to 0, the x max to 4, the y min to 0, and the y max to 100. Press graph and then press second trace and then press maximum to determine the maximum volume. The maximum volume is 90.7in3. (D) How big should the cut-out squares be in order to produce the box of maximum volume? In part C when finding the maximum volume, the calculator said that the maximum volume of 90.7 in3 was at x = 1.66. This means when the squares are cut out with a length of 1.66 inches, the volume is at it’s peak. The data in table 1.12 is low and high temperatures observed on 9/9/1999 in major American Cities. Find a function that approximates the high temperature (y) as a function off the low temperature(x). Use this function to predict the high temperature that day for Madison, WI, given that the low was 46. Plug the values table on page 156 into your calculator into L­1 and L2. Once the data has been entered, turn on one of the stat plots to a scatter plot and graph the points. Since the points are in a general positive pattern, we can use a linear regression to determine the equation. Under stat, scroll over to linear regression and press enter. The calculator says the equation is y= .97x + 23. Now press the y= button and graph the equation. The line follows the general trend of the graph. Now plug in 46(the low temperature) for x. The predicted high temperature for Madison WI according to the equation should be 67.62 degrees.
 * __ Chapter 1: Lesson 7  __**
 * Example 2 (pg 152) **
 * Example 6 (pg 156) **
 * Homework - pg 161 #’s 33 and 55 **