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Suppose you are given a piece of paper which is 9 in. x 11 in. and your goal is to make the piece of paper into an open-top box by removing a square from each corner of the box, folding up the flaps, and taping the flaps to the sides. Let's find what size square to cut to maximize the volume of the box!

Suppose you are given a piece of paper which is 9 in. x 11 in. and your goal is to make the piece of paper into an open-top box by removing a square from each corner of the box, folding up the flaps, and taping the flaps to the sides. Let's find what size square to cut to maximize the volume of the box!

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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the square that you cut out from each corner. e. Now let's generalize! Let x be the side length Find a formula for the volume V(x) of the box we make as a function of x. f. Realistically, note that we must have x > 0 in order to make a box. What value must x be less than in order to be able to make a box? Briefly explain your reasoning. g. Carefully simplify V (x) as much as you can, then find V'(x), locate the critical points, and make a sign chart for V'(x). Find the critical points exactly, but then use a calculator to estimate them to 2 decimal places (e.g. something like 2.34). (Recall we only care about the critical points in the physically relevant interval.) h. From your work in part (g), explain why we are guaranteed that V(x) has an absolute max (not just local) on the physically relevant interval. Where does the absolute max occur? Page 2 of 3
Transcribed Image Text:the square that you cut out from each corner. e. Now let's generalize! Let x be the side length Find a formula for the volume V(x) of the box we make as a function of x. f. Realistically, note that we must have x > 0 in order to make a box. What value must x be less than in order to be able to make a box? Briefly explain your reasoning. g. Carefully simplify V (x) as much as you can, then find V'(x), locate the critical points, and make a sign chart for V'(x). Find the critical points exactly, but then use a calculator to estimate them to 2 decimal places (e.g. something like 2.34). (Recall we only care about the critical points in the physically relevant interval.) h. From your work in part (g), explain why we are guaranteed that V(x) has an absolute max (not just local) on the physically relevant interval. Where does the absolute max occur? Page 2 of 3
This assignment is designed to give us pračtice tránšláting à physical šituation Ihto matH TU Inu the mil quantity (Problem 1), as well as practice computing antiderivatives and finding particular antiderivatives (Problem 2). е: Suppose you are given a piece of paper which is 9 in. x 11 in. and your goal is to make the piece of paper into an open-top box by removing a square from each corner of the box, folding up the flaps, and taping the flaps to the sides. Let's find what size square to cut to maximize the volume of the box! a. Suppose you decided to cut out squares 1 in. x 1 in. from each corner (see picture). What would be the volume of the box we make from it? 9 in. (Remember that the volume of a box of length l, width w, and height h is V = lwh.) 11 in. 11 in. . how long is this? ニ 63 ( V= 63 in 7.12 b. Now (from a fresh sheet of paper) you cut out squares 2 in. x 2 in. from each corner. What would the volume of the box be now? Draw your 9 in. own diagram this time. how long is this? c. What if you cut squares 3.5 in. x 3.5 in. from each corner? What is the volume now? d. Describe some observations from your work in parts (a), (b), (c). How does the volume change based on the size of the cut? Can you make an educated guess about what size squares would maximize the volume of the box? Page 1 of
Transcribed Image Text:This assignment is designed to give us pračtice tránšláting à physical šituation Ihto matH TU Inu the mil quantity (Problem 1), as well as practice computing antiderivatives and finding particular antiderivatives (Problem 2). е: Suppose you are given a piece of paper which is 9 in. x 11 in. and your goal is to make the piece of paper into an open-top box by removing a square from each corner of the box, folding up the flaps, and taping the flaps to the sides. Let's find what size square to cut to maximize the volume of the box! a. Suppose you decided to cut out squares 1 in. x 1 in. from each corner (see picture). What would be the volume of the box we make from it? 9 in. (Remember that the volume of a box of length l, width w, and height h is V = lwh.) 11 in. 11 in. . how long is this? ニ 63 ( V= 63 in 7.12 b. Now (from a fresh sheet of paper) you cut out squares 2 in. x 2 in. from each corner. What would the volume of the box be now? Draw your 9 in. own diagram this time. how long is this? c. What if you cut squares 3.5 in. x 3.5 in. from each corner? What is the volume now? d. Describe some observations from your work in parts (a), (b), (c). How does the volume change based on the size of the cut? Can you make an educated guess about what size squares would maximize the volume of the box? Page 1 of
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