a) the student has made multiple errors. Find at least three errors the student has madeB) write down the correct response the student should have given, showing all work We wish to construct the world's largest open-top cardboard box. We have exactly 800 squarefeet of cardboard to use, and the box we make must (due to Amazon regulations) be twice aswide as it is long. Of all the dimensions we could choose from, which box dimensions will resultin the box with the largest volume? List the dimensions for the bottom of the box, i.e., no needto find the final height of the box.STUDENT RESPONSE: Suppose the length of the box is c and its height is h. Then the sideshave area ch. Meanwhile, the width is 2x (twice the length), making the base of the box have area2x · x = 2x2. Thus, the surface area of the box isS = 4xh + 2x2Since we know the surface area will be 800 (the amount of material we have to work with), we canuse this to solve for h in terms of r:800 = 4xh +2x² → h =800 2x24xNow, the volume we wish to maximize isVolume = x · 2x · h = 2x²800-2x24xyielding our objective functionf (x) = 400x – x° & Domain = (0, 800)Finally, we wish to find the absolute maximum of f(x) (the maximum volume) by examining thesign of f"(x) = -6z. Since f"(x) goes from positive to negative at r = 0, the absolute maximummust be at I= 0.19tv

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Description: a) the student has made multiple errors. Find at least three errors the student has madeB) write down the correct response the student should have given, showing all work We wish to construct the world's largest open-top cardboard box. We have exactl

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Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter9: Systems Of Linear Equations

Section: Chapter Questions

Problem 10MPS

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a) the student has made multiple errors. Find at least three errors the student has made B) write down the correct response the student should have given, showing all work

We wish to construct the world's largest open-top cardboard box. We have exactly 800 square
feet of cardboard to use, and the box we make must (due to Amazon regulations) be twice as
wide as it is long. Of all the dimensions we could choose from, which box dimensions will result
in the box with the largest volume? List the dimensions for the bottom of the box, i.e., no need
to find the final height of the box.
STUDENT RESPONSE: Suppose the length of the box is c and its height is h. Then the sides
have area ch. Meanwhile, the width is 2x (twice the length), making the base of the box have area
2x · x = 2x2. Thus, the surface area of the box is
S = 4xh + 2x2
Since we know the surface area will be 800 (the amount of material we have to work with), we can
use this to solve for h in terms of r:
800 = 4xh +2x² → h =
800 2x2
4x
Now, the volume we wish to maximize is
Volume = x · 2x · h = 2x²
800-2x2
4x
yielding our objective function
f (x) = 400x – x° & Domain = (0, 800)
Finally, we wish to find the absolute maximum of f(x) (the maximum volume) by examining the
sign of f"(x) = -6z. Since f"(x) goes from positive to negative at r = 0, the absolute maximum
must be at I= 0.
19
tv

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