quantities and co-variation of quantities

Transcription

1 quantities and co-variation of quantities Module 2 : Investigation 1 MAT 170 Precalculus August 22, 2016

2 what is a quantity?

3 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. 3

4 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. 4

5 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. 5

6 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. 6

7 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. 7

8 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. 8

9 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 east wall 64 feet 1/3 9

10 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 10

11 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. 11

12 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 12

13 variables

14 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. 14

15 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? 15

16 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. 16

17 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...? 17

18 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) 18

19 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 full 2 gal. x The varying number of gallons of gasoline in your tank is given by 2 + x 19

20 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) Bill start Bob t 5 sec. t The varying number of seconds that Bob has been running is given by t

21 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) Bill start? t Bob 5 sec.? The varying number of seconds that Bill has been running is given by t 5. 21

22 question 9 The following graph relates the depth of water in a reservoir to the number of months since January 1, (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? 22

23 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, (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,

24 question 9 - possible answers The following graph relates the depth of water in a reservoir to the number of months since January 1, (b) Interpret the meaning of the point (4, 68). The water in the reservoir was 68 feet deep 4 months after January 1,

25 question 9 - possible answers The following graph relates the depth of water in a reservoir to the number of months since January 1, (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 = 27 feet. 25

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