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Topic #02 - Mathematical Relationships

Topic 2 - Concepts and Skills

Vocabulary, Variables and Equations

Review Questions

Application Questions

Word Problems






Topic #02 Concepts and Skills

Link to "Concepts and Skills"
 

Vocabulary, Variables and Equations

equation: an expression or a proposition, often algebraic, asserting the equality of two quantities

solution: the process of determining the answer to a problem

variable: a quantity or function that may assume any given value or set of values 

dependent variable: a variable in an experiment that depends on the value of another variable. As in y depends on x.

independent variable: a variable in an experiemnt that is not dependent on another variable, but whode value is often manipulted during the experiment.

controlled variable: variables that are held constant so they do not influence the ourcome of an experiment

data: values, such as observations and measurements, derived from scientific experiments

data table: a table of data organized in such a way to reveal patterns among observations and measurements 

graph: a series of points, discrete or continuous, as in forming a curve or surface, where each point represents a value of a given function.

function: a mathematical relationship expressed as an equation that describes a relationship between or among variable variable values

curve: a generic term used to mean the function / equation displayed on a graph

x-axis: the horizonatal line through the origin on a cartesian coordinate system where in 2 dimensions y = 0 and in 3 dimensions y = 0 and z = 0

y-axis: the vertical line through the origin on a cartesian coordinate system where in 2 dimensions x = 0 and in 3 dimensions x = 0 and z = 0

plot: to determine and mark (points), as on plotting paper (graph paper), by means of measurements or coordinates

data point: one point representing one piece of data on a graph. (On a two dimensional graph a data point is defined in terms of an x and y coordinate pair. On three dimensional graph a data point is represented by an x, y and z value.) 

slope: the rate of change in a derived quantity found by dividing the change in the y value divided by the change in the x value on a two dimensional graph

y-intercept: the y value in a function that has been plotted when the x value is zero

relationship: the mathematical connection between to variable values as defined by a function (equation)

direct variation: a realtionship that says that one variable such as the y variable varies directly with another variable such as the x variable. Mathematicians say this as "y is a function of x".

direct proportion: a means of saying that the variable values in a direct variation relationship are proportional to one another. That is as one increases so does the other.

inverse variation: a relationship that says that one variable such as the y variable varies inversely with another variable such as the x variable. Mathematicians say this as "y is a function of 1/x".

inverse proportion: a means of saying that the variable values in an inverse variation relationship are inversely proportional to one another. That is as one increases the other decreases proportionally.

linear relationship: another way of describig that a relationship between two variables is one of being a direct variation. Linear (functions) relationships always appear as straight lines when plotted on a two dimensional cartesian coordinate system.

quadratic relationship: a relationship that says that one variable such as the y variable varies directly with the square of another variable. Mathematicians say this as "y varying directly with the square of x is a parabolic relationship".
 

Review Questions:

Chapter 2
Reviewing Concepts

Questions:
Section 2.1
1. Why is SI important?
2. List the common SI base units.
3. How are base units and derived units related?
4. You convert the speed limit of an expressway given in miles per hour into meters per second and obtain the value 1.5 m/s.  Is this calculation likely to be correct? Explain.
5. Give the name for each multiple of the meter.
a. 1/100m
b. 1/1000m
c. 1000m
6. How may units be used to check on whether a conversion factor has been used correctly?

Section 2.2
7. What determines the precision of a measurement?
8. Explain how a measurement can be precise but not accurate.
9. How does the last digit differ form the other digits in a measurement?
10. Your lab partner recorded a measurement as 100g.
a. Why is it difficult to tell the number of significant digits in this measurement?
b. How can the number of significant digits in such a number be made clear?

Section 2.3
11. How do you find the slope of a linear graph?
12. A person who has recently consumed alcohol usually has longer reaction times than a person who has not.  Thus, the time between seeing a stoplight and hitting the brakes would be longer for the drinker than for the nondrinker.
a. For a fixed speed, would the reaction distance for a driver who had consumed alcohol be longer or shorter than for a nondrinking driver?
b. Would the slope of the graph of that reaction distance versus speed have the steeper or the more gradual slope?
13. During a laboratory experiment, the temperature of the gas in a balloon is varied and the volume of the balloon is measured.  Which quantity is the independent variable?  Which quantity is the dependent variable?
14. For a graph of the experiment in 13,
a. What quantity is plotted on the horizontal axis?
b. What quantity is plotted on the vertical axis?
15. A relationship between the independent variable x and the dependent variable y can e written using the equation y = ax^2, where a is a constant.
a. What is the shape of the graph of this equation?
b. If you define a quantity z = x^2, what would be the shape of the graph obtained by plotting y versus z?
16. Given the equation F = mv^2/R, what relationship exists between
a. F and R?
b. F and m?
c. F and v?
17. Based on the equation in problem 16, what type of graph would be drawn for 
a. F versus R?
b. F versus m?
c. F versus v?

 

Application Questions:

Chapter 2
Applying Concepts

Questions

18. the density of a substance is its mass per unit volume.
a. Give a possible metric unit for density.
b. Is the unit for density base or derived?
19. Use figure 2-4 to locate the size of the following objects.
a. the width of your thumb
b. the thickness of a page in this book.
c. the height of your classroom.
d. the distance from your home to your classroom.
20. Make a chart of sizes of objects similar to the one shown in figure 2-
4.  Include only objects that you have measured.  Some should be less than 
one millimeter; others should be several kilometers.
21. Make a chart similar to figure 2-4 of time intervals.  Include intervals 
like the time between heartbeats, the time between presidential elections, 
the average lifetime of a human, the age of the United States.  Find as many 
very short and very long examples as you can.
22. Three students use a meterstick to measure the width of a lab table. 
One records a measurement of 84cm, another of 83.8 cm, and the third of 
83.78cm.  Explain which answer is recorded correctly.
23. Two students measure the speed of light.  One obtains (3.001 +- 0.001) * 
10^8 m/s; the other obtains (2.999 += 0.006) * 10^8 m/s
a. Which is more precise
b. Which is more accurate
24. Why can quantities with different units never be added or subtracted but 
can be multiplied or divided? Give examples to support your answer.
25. Suppose you receive $5.00 at the beginning of a week and spend $1.00 
each day for lunch.  You prepare a graph of the amount you have left at the 
end of each day for one week.  Would you the slope of this graph be 
positive, zero or negative?  Why?
26. Data are plotted on a graph and the value on the y-axis is the same for 
each value of the independent variable. What is the slope?  Why?
27. The graph of braking distance versus car speed is part of a parabola. 
Thus, we write the equation d = av^2 + bv + c.  The distance, d, has units 
meters, and velocity, v, has units meter/second.  How could you find the 
units of a, b, and c?  What would they be?
28. In baseball, there is a relationship between the distance the ball is 
hit and the speed of the pitch.  The speed of the pitch is the independent 
variable.  Choose your own relationship.  Determine which is the independent 
variable and which is the dependent variable.  If you can, think of other 
possible independent variables for the same dependent variables.
29. Aristotle said that the quickness of a falling object varies inversely 
with the density of the medium through which it falls.
a. According to Aristotle, would a rock fall faster in water (density 1000 
kg/m^3) or in air (density 1kg/m^3)?
b. How fast would the rock fall in a vacuum?  Based on this, why would 
Aristotle say that there could be no such thing as a vacuum.

 

Word Problems:

Problem Set #2:  Follow the written directions unless indicated otherwise by your instructor.

1. You know that 2 = 8 / 4 .  Using the ideas that you have learned from studying algebra, rewrite this statement (rem: an equation is called a statement) such that the rest of the statement equals the number shown.

 a. 8 b. 4

2. Solve the following equations for v:

 a. d = v t  b. t = d / v  c. a = v2 / 2 s  d. v /a = b / c

3. Solve the equation d = a t2 / 2  for:

 a. t2  b. a  c. 2

4. solve for the variable E for:

 a. f = E / s b. m = 2E / v2  c. E / c2 = m

5. Solve the equation P = F v for:

 a. v  b. f 

6. solve the equation v2 = 2 d a for:

 a. d  b. a

7. solve each of these equations for x:

 a. w = f x b. g = f / x c. n = x / y d. d = a x2 /2

8. Find the answers to these problems using consistent units.

 a. Find the area of  a rectangle 2 mm by 30 cm.

 b. Find the perimeter of a rectangle 25 cm by 2.00 mm
 

9.  Substitute the suitable units into the following equations and state which are correct outcomes. Do write a brief explanation if an outcome is wrong, why it is wrong.

 a. area = length x width x height [use cm for length, width, and height]

 b. time = distance / speed [use m for distance, and m / s for speed]

 c distance = speed x  time2. [use m / s for speed and s for time]

Graphing functions and plotting data:

 Graph 1:  Linear graphs.  A set of x, y data pairs starts out at 0, 0 with y increasing at
  twice the rate of the x variable. Generate a data chart and graph of this data,
  if the second data pair is 1, 2, and there are 4 more pairs of data.

 Graph 2:  Hyperbolic graphs.  A set of data starts out at 1, 50 with y decreasing by
  half each time x is doubled. Generate a data chart and graph of this data, if the
  x value doubles five times.

 Graph 3:  Parabolic graphs.  A set of data starts out at 0, 0, with y increasing as the
  square of x. Generate a data chart and graph of this data, if the x value increases
  by a factor of one for the next six consecutive pairs of data.

The following section deals with triangles and trig functions.

10. Draw a triangle with a horizontal line (base line) labeled s. Construct a vertical line, the one at a right angle to the base line upwards from the base line at the right end of the base line. Label this vertical line r. Next, connect the far left end of the base line with the top of the vertical line. This third line is called the hypotenuse of the right triangle. Label this hypotenuse t. Finally, label the angle opposite side s as S, the angle opposite side r as R, and the angle opposite side t as T. Having done this, you should answer the following questions. 

 a. Which side of the triangle is opposite angle R?
 b. Which side is adjacent to angle R?
 c.  Write the equations for the sin, cos, and tan using angle R as your reference angle.

11.a.  Find the trignometric function values for the following angles:

 a. sin A, for  A = 10, 30, and 45 degrees
 b. cos A, for A = 10, 30, and 45 degrees
 c. tan A, for A = 10, 30, and 45 degrees

11.b  Find the size of the angles associated with each trigonometric function below.  Often the Greek letter theta, q , is used to designate the unknown angle. We are using the letter A, for angle, in this situation. [On the calculator use the inverse key or 2nd function key with the trig. function indicated to find the angle.]

 a  sin A = 0.500 b. sin A = 0.985 c. cos A = 0.707
 d. sin A = 0.707 e. tan A = 1.00  f. tan A =  0.364
 g. tan A = 2.050 h  cos A = 0.866

12.  One angle of a right triangle is 20.0 degrees.  The length of the Hypotenuse is 6 cm.
        a. draw the triangle to scale and measure the lengths of the other two sides
        b. use trig to calculate the lengths of these two sides.

13.  one angle of a right triangle is 35 degrees. the length of the opposite side is 14 cm.  Use the tangent of 35 degrees to calculate the length of the side adjacent to the angle.

14. If a baseball is hit at an angle of 14o, how high will it be after covering a horizontal distance of 84 m?  How far will it have traveled through the air? (You may assume a straight path.)