An open box of maximum volume is to be made from a square piece of material, s = 12 inches on a side, by cutting equal squares from the coners and turning up the sides (see figure). S-2r X (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Height, xLength and Width Volume, V 12 2(1) 1 1[12 2(1)12 2[12 2(2)12 = 100 12 2(2) 2. = 128 3[12 2(3)12 12 2(3) 3 12 2(4) 4[12 2(4)]2 4 12 2(5) 5[12 2(5)]2 12 2(6) 6[12 2(6)]2 Use the table to quess the maximum volume. =A (b) Write the volume Vas a function of x. 0 x
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Which is the result?
![An open box of maximum volume is to be made from a square piece of material, s = 12 inches on a side, by cutting equal squares from the
coners and turning up the sides (see figure).
S-2r
X
(a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.)
Height, xLength and
Width
Volume, V
12 2(1)
1
1[12 2(1)12
2[12 2(2)12
= 100
12 2(2)
2.
= 128
3[12 2(3)12
12 2(3)
3
12 2(4) 4[12 2(4)]2
4
12 2(5) 5[12 2(5)]2
12 2(6) 6[12 2(6)]2
Use the table to quess the maximum volume.
=A
(b) Write the volume Vas a function of x.
0 x<6
(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
120
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10
60
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X
2
4
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6
X
0.5
1.0
1.5
2.0
2.5
3.0
V
V
60
120
50
100
40
80
30
60
20
40
10
20
X
6
1
X
2
3
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5
3
5
2
S 2r](https://prod-qna-question-images.s3.amazonaws.com/question/0291f6f9-8c73-44e5-9b62-f3f476a1e310/24b85acc-632b-4f9c-9ddc-cb3dc9c2177d/8ztmbg8.jpeg)
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An open box of maximum volume is to be made from a square piece of material, s = 12 inches on a side, by cutting equal squares from the coners and turning up the sides (see figure). S-2r X (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Height, xLength and Width Volume, V 12 2(1) 1 1[12 2(1)12 2[12 2(2)12 = 100 12 2(2) 2. = 128 3[12 2(3)12 12 2(3) 3 12 2(4) 4[12 2(4)]2 4 12 2(5) 5[12 2(5)]2 12 2(6) 6[12 2(6)]2 Use the table to quess the maximum volume. =A (b) Write the volume Vas a function of x. 0 x<6 (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 120 15 100 80 10 60 40 20 X 2 4 5 6 X 0.5 1.0 1.5 2.0 2.5 3.0 V V 60 120 50 100 40 80 30 60 20 40 10 20 X 6 1 X 2 3 4 5 3 5 2 S 2r
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