- Home
- INTERMEDIATE ALGEBRA
- Course Syllabus for Algebra I
- Mid-Plains Community College
- FRACTION OF A WHOLE NUMBER
- Systems of Linear Equations
- MATH FIELD DAY
- Course Outline for Finite Mathematics
- Calculus
- Algebra Final Examination
- Math 310 Exam #2
- Review of Trigonometric Functions
- Math 118 Practice test
- Precalculus Review
- Section 12
- Literal Equations
- Calculus Term Definitions
- Math 327A Exercise 2
- Public Key Algorithms II
- Maximizing Triangle Area
- Precalculus I Review for Midterm
- REVIEW OF A FIRST COURSE IN LINEAR ALGEBRA
- Math 6310 Homework 5
- Some Proofs of the Existence of Irrational Numbers
- ALGEBRAIC PROPERTIES OF MATRIX OPERATIONS
- Math 142 - Chapter 2 Lecture Notes
- Math 112 syllabus
- Math 371 Problem Set
- Complex Numbers,Complex Functions and Contour Integrals
- APPLICATIONS OF LINEAR EQUATIONS
- Week 4 Math
- Fractions
- Investigating Liner Equations Using Graphing Calculator
- MATH 23 FINAL EXAM REVIEW
- Algebra 1
- PYTHAGOREAN THEOREM AND DISTANCE FORMULA
- Georgia Performance Standards Framework for Mathematics - Grade 6
- Intermediate Algebra
- Introduction to Fractions
- FACTORINGS OF QUADRATIC FUNCTIONS
- Elementary Algebra Syllabus
- Description of Mathematics
- Integration Review Solutions
- College Algebra - Applications
- A Tip Sheet on GREATEST COMMON FACTOR
- Syllabus for Elementary Algebra
- College Algebra II and Analytic Geometry
- Functions
- BASIC MATHEMATICS
- Quadratic Equations
- Language Arts, Math, Science, Social Studies, Char
- Fractions and Decimals
- ON SOLUTIONS OF LINEAR EQUATIONS
- Math 35 Practice Final
- Solving Equations
- Introduction to Symbolic Computation
- Course Syllabus for Math 935
- Fractions
- Fabulous Fractions
- Archimedean Property and Distribution of Q in R
- Algebra for Calculus
- Math112 Practice Test #2
- College Algebra and Trigonometry
- ALGEBRA 1A TASKS
- Description of Mathematics
- Simplifying Expressions
- Imaginary and Complex Numbers
- Building and Teaching a Math Enhancement
- Math Problems
- Algebra of Matrices Systems of Linear Equations
- Survey of Algebra
- Approximation of irrational numbers
- More about Quadratic Functions
- Long Division
- Algebraic Properties of Matrix Operation
- MATH 101 Intermediate Algebra
- Rational Number Project
- Departmental Syllabus for Finite Mathematics
- WRITTEN HOMEWORK ASSIGNMENT
- Description of Mathematics
- Rationalize Denominators
- Math Proficiency Placement Exam
- linear Equations
- Description of Mathematics & Statistics
- Systems of Linear Equations
- Algebraic Thinking
- Study Sheets - Decimals
- An Overview of Babylonian Mathematics
- Mathematics 115 - College Algebra
- Complex Numbers,Complex Functions and Contour Integrals
- Growing Circles
- Algebra II Course Curriculum
- The Natural Logarithmic Function: Integration
- Rational Expressions
- QUANTITATIVE METHODS
- Basic Facts about Rational Funct
- Statistics
- MAT 1033 FINAL WORKSHOP REVIEW
- Measurements Significant figures
- Pre-Calculus 1
- Compositions and Inverses of Functions

# Math Problems

**Mathematics Contained in these Three Lessons
**"Reading This Could Help You Sleep: Caffeine in Your Body" is an
introduction to

exponential functions of the form at the Intermediate Algebra level, with

emphasis on the meaning of these functions and their graphs. The concept of half-life is

introduced. A conditional function (a piecewise-defined function) is used.

"Get the Lead Out" extends the study of exponential functions and can be used to

introduce the use of logarithms to "un-do" exponential expressions in solving equations.

"So Much Coffee, So Little Time" can be used at the Intermediate Algebra level to

help students see the value of "solving systems of linear equations". It shows connections

between solving linear equations and exponential functions, and continues the study of

exponential functions this time of the form . End behavior of these

functions with horizontal asymptotes is included.

**Set-up**

Each students should have a graphing calculator. Alternatively, each group of 4

students should work at one computer.

**Organization**

These three lessons are interdependent. They should be done in the ordered listed

above in the title to the Teacher Materials. You can do the first without doing the second

or the third, or you can do the first and second without doing the third. We recommend

that you not try to use the second or third unless you have done the one(s) that precede it.

Each unit can be used at a time that is appropriate for the course, that is, they do not need

to be covered on consecutive class days.

**Answers to Problems and Teaching Suggestions for Reading This Could Help You**

Sleep: Caffeine in Your Body

Sleep: Caffeine in Your Body

**Reflection Point:**Students should have attempted problem 1 for homework. You can

begin class by discussing the reading and the students' answers. When you discuss this

homework problem in class, have students describe how they actually computed the

answers for part a. Be sure that all students see the relationship between multiplying

again by 0.87 and increasing the exponent on 0.87 by 1. Having different students

explain their reasoning for problem 1b in class should help all students understand the

process. After the discussion, hand out

**Eliminating Caffeine from**

**the Body,**page 3 of

this activity, and have the students work on this page in groups of 3 or 4.

c. As t gets large,

d. This point is at about and is shown with
a box in the graph.

e. When . This point is shown with a + on the
graph.

Students can get this result approximately from their graphs. You might wish to
remind

students that an exponent of indicates a
square root. If so, they can calculate

and compare with their reading from the
graph.

f. When .

g. After two half-lives, , which is
;after three half-lives,
,

half of the previous result hours is two
half-lives; three half-lives is

approximately hours.

Students can get these results approximately from their graphs. Help them
recognize that

after each 5 hour period (approximately), the amount of caffeine remaining in
the body is

about half what it was at the beginning of the 5-hour period. Studying the graph
with

this in mind will help to prepare students for later work with half-life. These
three points

are marked with × on the graph.

**Homework**

Problem 3 is intended to extend students' familiarity with exponential
functions.

Each person's table and graph will differ from the others, of course, but we
suppose most

will have more than one drink during the day containing caffeine. In this case,
most will

have to write a conditional function. For a person who consumes130 mg at 8:00
am, 40

mg at 9:30 am, and 165 mg at 1:00 pm, the conditional function could be written
as

or as

where t measures time in hours after 8 am. Since the
person in our table had her first cup

of coffee at 8 am, she measures t in hours since 8 am. Thus, after 9:30 am, she
will need

to use as an exponent to indicate hours that
have passed since her 9 am Dr.

Pepper and later, as an exponent to indicate
hours that have passed since her 1pm

espresso. Note that in the first form of the function, the total amount of
caffeine in the

person present at time and
was computed and used to start the next

exponential. In the second form of the function, the caffeine in each drink is
considered

as being eliminated exponentially and independently from the caffeine in the
other drinks.

To graph such a function on a graphing calculator, set the calculator to DOT
mode

(as opposed to CONNECTED) to avoid lines joining endpoints. The function can be

graphed several different ways. For example, students could graph

or

or

with the window set for and
with max depending on the

student. The graph should look something like the following graph.

**Answers to Problems and Teaching Suggestions for Get
the Lead Out
**1. a. 0.9850, 0.9702, 0.9557, 0.9413 mg, respectively

b. 7 days

c.

**It is assumed students will have worked problem 1 for homework. If**

Reflection Point:

Reflection Point:

you discuss this homework problem in class, ask students how they did the problems.

You might then note that the answer to part b is the approximate solution to the equation

and discuss how you could use this equation to find t; that is, take the log

of both sides. If you plan to teach solution of exponential equations "by hand", now is a

good time to help students recognize the need for an inverse for exponential functions

with which to do it. The equations can be solved using a graphing utility, of course, but if

students try that method first, they will be ready to recognize that the search for an

appropriate window for a solution to these equations may not be as fast or as easy as a

logarithmic approach.

After discussing problem 1, you should hand out

**Get the Lead Out**Classroom

materials and have students work on problems 2 and 3 in groups of 3 or 4. You can also

have them work on problem 4 on page 3 of this activity, or you can assign that problem

for homework.

2. a. 1.9700, 1.9404, 1.9113 mg, respectively

b. After 46 days (or 45.862), the amount of lead in her body will be

reduced to 1 mg.

If students have any difficulty answering part b, ask them to explain how they worked

part a. Remind them about the caffeine activity. Help them generalize for any and then

recognize that the function is exactly that generalization. The determination of half-life

can be found from the graph or by taking the log of both sides of the equation

Discussion to review the term half-life would be helpful here.

c. Another 46 days.

Here you might also show students another form of the function based on the half-life

idea: It might help them understand this if they use their calculator to

see that

d. The half-life is 45.862 days. This means that every 45.862 days, half of the lead is

eliminated from the blood.

e. To a level of 0.4 mg: 106.5 days. To a level of 0.2 mg: another 46 days, or 45.862 -

thus about 152 days.

For the 0.4 mg question, students might use their graphing calculators or they might

calculate using logarithms. To answer the 0.2 mg question, they have many choices,

but we hope they will use the half-life idea of part d.

3. one hundredth of
one percent is Be aware of the

potential for error here. Misunderstanding of this decimal will lead students
far astray

on the rest of the problem!

a.

b. 18.989 or 19 years; 37.9788 or 38 years

c. 18.989 or 19 years

**Reflection point:** This is a good time to ascertain that students
understand half-life.

4. a. If 0.02% of the cadmium is removed from the body each day, it will take
about

3465 days, or 9.49 years. The easiest way to find this is to solve
.If

0.004 % of the cadmium is removed from the body each day, it will take about

17328 days, or 47.47 years, which can be found by solving
So the

half-life for cadmium is between 9.5 and 47.5 years, depending on the person.

**Answers to Problems and Teaching Suggestions for So
Much Coffee, So Little Time:
More on Caffeine in the Body**

This activity can be used to help students see the value of "solving systems of

linear equations" at the Intermediate Algebra level. It shows connections between solving

linear equations and exponential functions; that is, we need to use a number of different

mathematical concepts to analyze real problems.

1. After n cups, t hours after the first cup, there will be f(t) grams of caffeine in the

body.

n cups | hours | mg of caffeine in body |

2. Solve to get
and
Graph

It will be seen from the graph that this function is monotone increasing and its
end

behavior is to approach the line , although
students may not recognize

immediately that the asymptote is exactly the constant term of the function.
From this

information, students can conclude that no one will die of an overdose of
caffeine, which

would happen at a level of about 5000 mg, by drinking 1 cup of coffee per hour.

**
Reflection point:** Students should have attempted problems 1 and 2 for
homework. At

the beginning of class, initiate a whole-class discussion to be sure students understand the

form of this exponential function, , why this one (with ) does

not go off to infinity, and how they might have known what its horizontal asymptote

would be from looking at the equation. One result of this example is that the amount of

caffeine stabilizes at an equilibrium value of 769.23 mg in the body. Discuss this result

with students in relation to medicines which are given periodically; that is, that the

amount of medicine in the body stabilizes at a level deemed appropriate by the physician.

Pass out

**Smokin'**and have the students work on problems 3, 4, and 5 in groups.

3. a. Solve to get
and .
Thus, the function

is

Assure that students see this result geometrically (on
their graphs) and algebraically

(from studying the function). It would also be valuable to see it numerically
from an

electronically generated table.

4. a. The amount absorbed each year is
, so .

Also, , so students need to solve the
equations

giving and
, so and

mg of cadmium.

b. The amount absorbed each year is , so

Also, , so students need to solve the
equations

giving and
so and

mg of cadmium.

5. 47.7 (our result in problem 3) is about 1.6 times the "average." In order to
assess the

relationship between our result that the smoker receives 47.7 mg more cadmium as
a

result of smoking and the "average" amount of 30 mg, we would need to know what

proportion of the 50 year old population has smoked for 30 years and what is the

average number of cigarettes smoked per day. We would also need to know for how

long and at what ages other fractions of this population smoked. That the added

amount due to cigarettes is actually between about 10 mg and about 150 mg makes
it

even harder to assess the validity of this statement. There is no right or wrong
answer

to this problem. Hopefully it will cause students to think more deeply about the
math

and about what additional information might be needed before making a statement
like

that given in this problem.