Packaging. Refer to Problems 1 and 2.
(A) Examine the graph of V(x) from Problem 1D and discuss the possible locations of other values of x that would produce a box with a volume of 65 cu in.
(B) Construct a table like Table 1 to estimate any such value to one decimal place.
(C) Refine the table you constructed in part (B) to provide an approximation to two decimal places.
Table 1 Volume
x | V(x ) |
1.1 | 62.524 |
1.2 | 64.512 |
1.3 | 65.988 |
1.4 | 66.976 |
1.5 | 67.5 |
1.6 | 67.584 |
1.7 | 67.252 |
Problem 1
Packaging. A candy box will be made out of a piece of cardboard that measures 8 by 12 in. Equal-sized squares x inches on a side will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box.
(A) Express the volume of the box V(x) in terms of x.
(B) What is the domain of the function v (determined by the physical restrictions)?
(C) Complete Table 2.
Table 2 Volume
x
V(x)
1
2
3
(D) Plot the points in part (C) and sketch a graph of the volume function using these points.
Problem 2
Packaging. Refer to Problem 3.
(A) Table 3 shows the volume of the box for some values of x between 1 and 2. Use these values to estimate to one decimal place the value of x between 1 and 2 that would produce a box with a volume of 65 cu in.
Table 3 Volume
x
V(x )
1.1
62.524
1.2
64.512
1.3
65.988
1.4
66.976
1.5
67.5
1.6
67.584
1.7
67.252
(B) Describe how you could refine this table to estimate x to two decimal places.
(C) Carry out the refinement you described in part (B) and approximate x to two decimal places.
Problem 3
Packaging. A candy box will be made out of a piece of cardboard that measures 8 by 12 in. Equal-sized squares x inches on a side will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box.
(A) Express the volume of the box V(x) in terms of x.
(B) What is the domain of the function v (determined by the physical restrictions)?
(C) Complete Table 4.
Table 4 Volume
x
V(x)
1
2
3
(D) Plot the points in part (C) and sketch a graph of the volume function using these points.
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