Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let x represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume V of the box as a function of x . Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that yield a maximum volume. (d) Find values of x such that V = 56 . Which of these values is a physical impossibility in the construction of the box? Explain.
Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let x represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume V of the box as a function of x . Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that yield a maximum volume. (d) Find values of x such that V = 56 . Which of these values is a physical impossibility in the construction of the box? Explain.
Solution Summary: The author illustrates the diagram showing the squares removed from a rectangular piece of material with dimensions 15 centimetres and 9cm and the resulting dimensions of the open box.
Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides.
(a) Let
x
represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box.
(b) Use the diagram to write the volume
V
of the box as a function of
x
. Determine the domain of the function.
(c) Sketch the graph of the function and approximate the dimensions of the box that yield a maximum volume.
(d) Find values of
x
such that
V
=
56
. Which of these values is a physical impossibility in the construction of the box? Explain.
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