Maximum Volume You construct an open box from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure). (a) Write a function V that represents the volume of the box. (b) Determine the domain of the function V . (c) Use a graphing utility to construct a table that shows the box heights x and the corresponding volumes V ( x ) . Use the table to estimate the dimensions that produce a maximum volume. (d) Use the graphing utility to graph V and use the graph to estimate the value of x for which V ( x ) is a maximum. Compare your result with that of part (c).
Maximum Volume You construct an open box from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure). (a) Write a function V that represents the volume of the box. (b) Determine the domain of the function V . (c) Use a graphing utility to construct a table that shows the box heights x and the corresponding volumes V ( x ) . Use the table to estimate the dimensions that produce a maximum volume. (d) Use the graphing utility to graph V and use the graph to estimate the value of x for which V ( x ) is a maximum. Compare your result with that of part (c).
Solution Summary: The author explains the function V that represents the volume of the box, made from a square piece of material, 36 inches, by cutting equal squares with sides of length x from the corners.
Maximum Volume You construct an open box from a square piece of material,
36
inches on a side, by cutting equal squares with sides of length
x
from the corners and turning up the sides (see figure).
(a) Write a function
V
that represents the volume of the box.
(b) Determine the domain of the function
V
.
(c) Use a graphing utility to construct a table that shows the box heights
x
and the corresponding volumes
V
(
x
)
. Use the table to estimate the dimensions that produce a maximum volume.
(d) Use the graphing utility to graph
V
and use the graph to estimate the value of
x
for which
V
(
x
)
is a maximum. Compare your result with that of part (c).
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