quantities and co-variation of quantities
Module 2 : Investigation 1
MAT 170 | Precalculus
August 22, 2016
what is a quantity ?
question 1
Consider a chair in our classroom :
(a) Identify 3 attributes of the chair that can be measured.
(b) Which of the attributes from part (a) have fixed values ?
(c) Which of the attributes from part (a) have varying values ?
(d) Identify 3 attributes of the chair that cannot be measured.
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question 1 - possible answers
(a) Identify 3 attributes of the chair that can be measured.
1. Height from the bottom of the rollers to the top of the back.
2. Weight of the chair.
3. Number of legs.
(b) Which of the attributes from part (a) have fixed values ?
1. Weight of the chair.
2. Number of legs.
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question 1 - possible answers
(c) Which of the attributes from part (a) have varying values ? How
are you determining this value ?
The height from the bottom of the rollers to the top of the seat
back will vary, because the height is adjustable.
(d) Identify 3 attributes of the chair that cannot be measured.
1. How comfortable the chair is.
2. How the chair smells.
3. The style of the chair.
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definition of a quantity
Definition
The attributes of an object that can be measured are called
quantities. To clearly describe a quantity, we must specify three
things :
1. The object being measured,
2. the attribute of the object that is being measured,
3. and the units used in the measurement.
If the value of a quantity does not change, we say that the
quantity is a fixed quantity and its value is a constant.
If the value of a quantity does change, we call it a varying quantity.
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question 1 (continued)
Back to our chair... Imagine the chair moving from the east wall to
the west wall of the classroom.
(e) Is the chair’s distance (in feet) from the east wall a quantity ? If
so, is it fixed or varying ?
(f1) Does the distance between the two walls vary or is this distance
constant/non-varying ?
(f2) Suppose the distance across the classroom from the east wall to
the west wall is 64 feet. Draw a diagram that represents this distance.
(g) Suppose you push the chair from the east wall straight across the
room to the west wall. In your diagram from part (f2), illustrate the
location of the chair when it has moved 1/3 of the way across the
classroom from the east wall.
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question 1 (continued) - possible answers
(e) Is the chair’s distance (in feet) from the east wall a quantity ? If
so, is it fixed or varying ?
Yes, it is a varying quantity since the chair can be moved towards
or way from the east wall.
(f1) Does the distance between the two walls vary or is this distance
constant/non-varying ?
That distance is (hopefully) non-varying.
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question 1 (continued) - possible answers
(f2) Suppose the distance across the classroom from the east wall to
the west wall is 64 feet. Draw a diagram that represents this distance.
west wall
east wall
64 feet
(g) Suppose you push the chair from the east wall straight across the
room to the west wall. In your diagram from part (f2), illustrate the
location of the chair when it has moved 1/3 of the way across the
classroom from the east wall.
west wall
our chair
64 feet
east wall
1/3
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question 1 (continued)
Still thinking about our chair...
(i) What two quantities would you use to compute the chair’s
distance (in feet) to the west wall ?
(j) Describe the chair’s distance (in feet) from the west wall in terms
of the chair’s distance from the east wall by using words to complete
the following statement :
The chair’s distance (in feet) from the west wall is
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question 1 (continued) - possible answers
(i) What two quantities would you use to compute the chair’s
distance (in feet) to the west wall ?
We considered two quantities :
∙ The distance (in feet) between the east and west wall
(constant).
∙ The distance (in feet) between the east and the chair
(varying).
(j) Describe the chair’s distance (in feet) from the west wall in terms
of the chair’s distance from the east wall by using words to complete
the following statement :
The chair’s distance (in feet) from the west wall is
... the distance (in feet) between the east wall and the west wall
minus the chair’s distance (in feet) from the east wall.
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question 2 - possible answers
(a) Imagine an empty bath tub. You turn on the water and it begins
to fill. Identify a couple of quantities associated with this situation.
(Be specific when describing how the attribute is being measured,
including where it is being measured from.)
∙ Object - time
∙ Attribute - time elapsed since the water was turned on
∙ Units - minutes
∙ Varying or non-varying ? - varying
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variables
definition of a variable
Definition
A variable is a letter or symbol that is designated to represent
all possible values that a varying quantity can assume.
For example, the amount of time (in minutes) from now to the end of
class is a varying quantity.
Rather than write that description of the quantity each time we want
to reference it, we can define a variable :
Let t denote the amount of time (in minutes) from now to the
end of class.
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question 4
(a) Why is it useful to use a variable (symbol) to representing all the
values that a quantity can assume.
(b) Define variables to represent the values of the quantities you
defined in Question 2 (the “bathtub question”).
(c) When defining a variable, why is it important to describe where
the quantity is being measured from ?
(d) When defining a variable, why is it important to include the units
that are being used in measuring the quantity ?
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question 4 - possible answers
(a) Why is it useful to use a variable (symbol) to representing all the
values that a quantity can assume.
Describing a quantity using words over and over again is tedious !
(b) Define variables to represent the values of the quantities you
defined in Question 2 (the “bathtub question”).
Let t denote the time elapsed (in minutes) since the water was
turned on.
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question 4 - possible answers
(c) When defining a variable, why is it important to describe where
the quantity is being measured from ?
To remove ambiguity. For example, if I state that my quantity is
the temperature (in degrees Fahrenheit) of water in the tub, there
are several reasons this ambiguous :
∙ When are you taking the temperature of the water ?
∙ Where are you taking the temperature in the tub ?
(d) When defining a variable, why is it important to include the units
that are being used in measuring the quantity ?
To remove ambiguity.If we just have t = 2, does that mean 2 seconds, minutes, hours, months, etc... ?
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question 7
(a) Your gas tank has 2 gallons of gas remaining when you pull into a
gas station. Write an expression to represent the varying number of
gallons of gasoline in your tank in terms of x, the varying number of
gallons of gasoline that you add to the tank (Illustrate this situation
with a diagram first)
(d) Bob starts running. Bill starts running 5 seconds later. Write an
expression to represent the varying number of seconds Bob has
been running in terms of t, the varying number of seconds that Bill
has been running. (Illustrate this situation with a diagram first)
(e) Bob starts running. Bill starts running 5 seconds later. Write an
expression to represent the varying number of seconds Bill has been
running in terms of t, the varying number of seconds that Bob has
been running. (Illustrate this situation with a diagram first)
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question 7
(a) Your gas tank has 2 gallons of gas remaining when you pull into a
gas station. Write an expression to represent the varying number of
gallons of gasoline in your tank in terms of x, the varying number of
gallons of gasoline that you add to the tank (Illustrate this situation
with a diagram first)
empty
2 gal.
full
x
The varying number of gallons of gasoline in your tank is given
by
2+x
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question 7
(d) Bob starts running. Bill starts running 5 seconds later. Write an
expression to represent the varying number of seconds Bob has
been running in terms of t, the varying number of seconds that Bill
has been running. (Illustrate this situation with a diagram first)
start
Bill
t
Bob
5 sec.
t
The varying number of seconds that Bob has been running is
given by
t + 5.
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question 7
(e) Bob starts running. Bill starts running 5 seconds later. Write an
expression to represent the varying number of seconds Bill has been
running in terms of t, the varying number of seconds that Bob has
been running. (Illustrate this situation with a diagram first)
start
Bill
?
t
Bob
5 sec.
?
The varying number of seconds that Bill has been running is given
by t − 5.
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question 9
The following graph relates the depth of water in a reservoir to the
number of months since January 1, 1990.
(a) Define variables to represent
the values of the varying
quantities in this situation.
(b) Interpret the meaning of the
point (4, 68).
(g) How does the depth of the
water in the reservoir vary as the
number of months since January
1, 1990 increases from 4 to 7
months ?
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question 9 - possible answers
The following graph relates the depth of water (in feet) in a reservoir
to the number of months since January 1, 1990.
(a) Define variables to represent
the values of the varying
quantities in this situation.
Let d denote the depth of
water (in feet) in the reservoir.
Let m denote the number of
months since January 1, 1990.
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question 9 - possible answers
The following graph relates the depth of water in a reservoir to the
number of months since January 1, 1990.
(b) Interpret the meaning of the
point (4, 68).
The water in the reservoir
was 68 feet deep 4 months
after January 1, 1990.
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question 9 - possible answers
The following graph relates the depth of water in a reservoir to the
number of months since January 1, 1990.
(g) How does the depth of the
water in the reservoir vary as the
number of months since January
1, 1990 increases from 4 to 7
months ?
As the number of months after January 1, 1990 increases
from 4 to 7, the water in the
reservoir decreases by 68 −
41 = 27 feet.
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