the box problem and function notation
Module 3 : Investigation 1
MAT 170 | Precalculus
September 2, 2016
the box problem (question 1)
Consider what is involved in building a box (without a top) from an
8.5” by 11” sheet of paper by cutting squares from each corner and
folding up the sides.
(c) Which quantities in this situation vary ? Which quantities are
fixed ?
(d) Describe how the configuration of the box changes as the length
of the of the sided of the square cutout varies.
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the box problem (question 1) - possible answers
(c) Which quantities in this situation vary ? Which quantities are
fixed ?
Varying : Length of the side of the cutout square (in inches), length
of the box (in inches), width of the box (in inches)
Fixed : Length and width of the paper (in inches), weight of the
paper (in ounces)
(d) Describe how the configuration of the box changes as the length
of the of the sided of the square cutout varies.
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the box problem (question 1)
Consider what is involved in building a box (without a top) from an
8.5” by 11” sheet of paper by cutting squares from each corner and
folding up the sides.
(e) How does the volume of the box vary as the length of the side of
the cutout varies from 0 to 4.25 inches ? Animation
(f) Sketch of graph of the volume of the box (in cubic inches) with
respect to the length of the side of the square cutout (in inches).
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the box problem (question 1) - possible answers
(e) How does the volume of the box vary as the length of the side of
the cutout varies from 0 to 4.75 inches ? Animation
3
3
3
It increases from 0 in to approximately 66 in , then back to 0 in .
(f) Sketch of graph of the volume of the box (in cubic inches) with
respect to the length of the side of the square cutout (in inches).
75
50
25
-0.4
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
4.4
4.8
5.2
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question 2
Let x represent the varying length of the side of the square cutout in
inches. Let w represent the varying width of the box’s base in inches.
Let ℓ represent the varying length of the box’s base in inches.
x
w
l
(a) Complete the following table :
x
w
ℓ
V
0
2.25
4.25
5
NA
NA
11
6.5
2.5
NA
15
2
8.5
4
0
NA
NA
NA
0
58.5
0
NA
NA
NA
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question 2
Let x represent the varying length of the side of the square cutout in
inches. Let w represent the varying width of the box’s base in inches.
Let ℓ represent the varying length of the box’s base in inches.
x
w
l
(c) Define a formula for w in
terms of x.
(d) Define a formula for ℓ in
terms of x.
(e) Define a formula for V in
terms of x.
(g) Use your formula from part (e) to represent the volume of the
box when x = 3.
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question 2 - solutions
x
w
(c) Define a formula for w in
terms of x.
w = 11 − 2x
l
(d) Define a formula for ℓ in
terms of x.
ℓ = 8.5 − 2x
(e) Define a formula for V in terms of x.
V = ℓ · w · x = (8.5 − 2x)(11 − 2x)x
(g) Use your formula from part (e) to represent the volume of the
box when x = 3.
V = (8.5 − 2 · 3)(11 − 2 · 3)(3) = 37.5
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what is a function ?
The relationship between the volume (in cubic inches) of the box
and the length (in inches) of the side of the cutout square defines a
function.
Definition : Function
A function consists of three parts :
(1) A set of input values, called the domain of the function,
(2) A set of output values, called the range of the function,
(3) A rule that assigns each input value to exactly one output
value.
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what is a function ?
The most salient feature of a function is that the rule assigns each
input to exactly one output.
rule
domain
range
OK!
10
what is a function ?
The most salient feature of a function is that the rule assigns each
input to exactly one output.
rule
domain
range
OK!
11
what is a function ?
The most salient feature of a function is that the rule assigns each
input to exactly one output.
rule
domain
range
NOT OK!
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what is a function ?
In the case of the volume (in cubic inches) of the box in terms of the
length (in inches) of the side of the cutout square :
Rule :
x 7→ x(8.5 − 2x)(11 − 2x)
|{z}
|
{z
}
input
output
Domain : all values of x between 0 and 4.25
Range : all possible volumes
Note that each x is assigned to one and only one volume.
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function notation
Let’s once again consider the function giving the volume of the box
(in cubic inches) in terms of the length (in inches) of the side of the
cutout square.
Let’s name our function V so that we remember that the output is
volume.
rule
name input
z
}|
{
z}|{ z}|{
V ( x ) = x(8.5 − 2x)(11 − 2x)
|
{z
}
function definition
So rather than writing :
When x = 3, the volume is 45.
We can simply write
V(3) = 45.
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question 3 & 5
Given that V(x) = x(11 − 2x)(8.5 − 2x) :
3(e) What is the smallest possible value for x ?
3(f) What is the largest possible value for x ?
3(g) What is the domain of V ?
5(a) Use function notation to represent the volume of the box when
the length of the side of the square cutout is 1.1 inches.
5(b) Use function notation to represent the volume of the box when
the length of the side of the square cutout is 2.5 inches.
5(c) Use function notation to represent the volume of the box when
the length of the side of the square cutout is c inches.
5(d) Use function notation to represent that the volume of the box is
62.5 cubic inches when the length of the side of the square cutout is
b inches.
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question 3 & 5 - solutions
Given that V(x) = x(11 − 2x)(8.5 − 2x) :
3(e) What is the smallest possible value for x ?
We must cutout an non-negative amount, so the smallest x can
be is 0.
3(f) What is the largest possible value for x ?
The original length of the paper is 8.5 inches, so the length of the
cutout can be at most half of length, 4.25 inches.
3(g) What is the domain of V ?
The value of x must be between 0 and 4.25 inches, so the domain
of V is [0, 4.25].
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question 3 & 5 - solutions
5(a) Use function notation to represent the volume of the box when
the length of the side of the square cutout is 1.1 inches.
V(1.1)
5(b) Use function notation to represent the volume of the box when
the length of the side of the square cutout is 2.5 inches.
V(2.5)
5(c) Use function notation to represent the volume of the box when
the length of the side of the square cutout is c inches.
V(c)
5(d) Use function notation to represent that the volume of the box is
62.5 cubic inches when the length of the side of the square cutout is
b inches.
V(b) = 62.5
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question 3 & 5
Given that V(x) = x(11 − 2x)(8.5 − 2x) :
5(e) Using your graphing calculator, solve for the length of the side
of the square cutout (x), when the volume is 62.5 cubic inches.
5(f) Using your graphing calculator, find an approximate value for the
maximum volume of the box.
5(g) Using your graphing calculator, solve for the length of the side of
the square cutout (x), when the volume is the maximum value you
found in (f ).
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question 3 & 5 - possible solutions
Given that V(x) = x(11 − 2x)(8.5 − 2x) :
5(e) Using your graphing calculator, solve for the length of the side
of the square cutout (x), when the volume is 62.5 cubic inches.
x ≈ 1.17462
and
x ≈ 2.03334
5(f) Using your graphing calculator, find an approximate value for the
maximum volume of the box.
maximum volume ≈ 66.1482
5(g) Using your graphing calculator, solve for the length of the side of
the square cutout (x), when the volume is the maximum value you
found in (f ).
x ≈ 1.58542
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