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Social Sciencea side, by cutting equal squares from theAn open box of maximum volume is to be made from a square piece of material, s = 36 inches oncorners and turning up the sides (see figure).S-2r-TXshown.)(a) Analytically complete six rows of a table such as the one below. (The first two rows areLength andWidthVolume, VHeight, x1[36 2(1)]22(1)11563612[36 2(2)]22(2)= 2048363[36 2(3)]236 2(3)34[36 2(4)]2 =36 2(4)2(5) 5[36 2(5)]2 =3636 2(6) 6[36 2(6)12 -Use the table to guess the maximum volume.V=(b) Write the volume V as a function of x.0 x< 18V=c) Use calculus to find the critical number of the function in part (b) and find the maximum value.V =(d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.VV350040030002500300200020015001000100500XX154102VV35001500300025001000200015005001000500X15101510st
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![a side, by cutting equal squares from the
An open box of maximum volume is to be made from a square piece of material, s = 36 inches on
corners and turning up the sides (see figure).
S-2r-
T
X
shown.)
(a) Analytically complete six rows of a table such as the one below. (The first two rows are
Length and
Width
Volume, V
Height, x
1[36 2(1)]2
2(1)
1156
36
1
2[36 2(2)]2
2(2)
= 2048
36
3[36 2(3)]2
36 2(3)
3
4[36 2(4)]2 =
36 2(4)
2(5) 5[36 2(5)]2 =
36
36 2(6) 6[36 2(6)12 -
Use the table to guess the maximum volume.
V=
(b) Write the volume V as a function of x.
0 x< 18
V=
c) Use calculus to find the critical number of the function in part (b) and find the maximum value.
V =
(d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.
V
V
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200
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100
500
X
X
15
4
10
2
V
V
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st](https://prod-qna-question-images.s3.amazonaws.com/question/0291f6f9-8c73-44e5-9b62-f3f476a1e310/aaf299fd-b433-470e-b1f8-628441eae1ba/pl8fj9c.jpeg)
Image Transcriptionclose
a side, by cutting equal squares from the An open box of maximum volume is to be made from a square piece of material, s = 36 inches on corners and turning up the sides (see figure). S-2r- T X shown.) (a) Analytically complete six rows of a table such as the one below. (The first two rows are Length and Width Volume, V Height, x 1[36 2(1)]2 2(1) 1156 36 1 2[36 2(2)]2 2(2) = 2048 36 3[36 2(3)]2 36 2(3) 3 4[36 2(4)]2 = 36 2(4) 2(5) 5[36 2(5)]2 = 36 36 2(6) 6[36 2(6)12 - Use the table to guess the maximum volume. V= (b) Write the volume V as a function of x. 0 x< 18 V= c) Use calculus to find the critical number of the function in part (b) and find the maximum value. V = (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph. V V 3500 400 3000 2500 300 2000 200 1500 1000 100 500 X X 15 4 10 2 V V 3500 1500 3000 2500 1000 2000 1500 500 1000 500 X 15 10 15 10 st
Expert Answer
(a)
Complete the table below analytically:
Image Transcriptionclose
Height, xLength and width 36-2(1 Volume, V 36-2(1) 341156 236-2(2)]=2[32= 2048 3[36-2(3)]3[30= 2700 4 36-2(4)283136 S[36-2(5)][263380 6[36-2(6)]24 3456 1 2 36-2(2) 3 36-2(3) 4 36-2(4) 5 36-2(5) 6 36-2(6) re
From the table, the maximum volume is 3456 cubic units.
(b)
As length and width of box is (s-2x ) = (36-2x ) each and height is x...
Image Transcriptionclose
V lengthxwidthxheight (36-2r)(36-2r)x 1296+4x2-144x\x V (x)4x3-144r2+1296x
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