Section 1.2
College Algebra
Mr. Faullin
Constructing Models to Solve Problems
Begin Class (10 minutes)
Attendance
Check in to class on your iPad.
Log Entry
Please update your time log.
Warm Up Problem
Solve for y.
Ax + By =
C
Submit your answer in the ‘Short Answer’ box on the Student Response webpage.
Break
Please put all electronic devices away at this time.
Lecture (20 minutes)
Definition – A formula is
Solving for a Variable in a Formula
Solve formulas using the same procedure for solving equations. Treat all variables that are NOT
being solved for as if they are just numbers.
Example – Solve for x.
1 1 1
=
+
y x w
Models
Definition – A model is
Example – Jeff knows that his neighbor Sarah paid $40,230, including sales tax, for a new car. If the sales
tax rate is 8%, then what is the cost of the car before tax?
Word Problems
Strategy
1. Read the problem until you understand the problem.
2. Draw a picture if possible.
3. Decide what the variable should represent. Be specific! (Best Advice Ever: Let the variable be
the thing you know the least about.)
4. Define all other unknowns in terms the variable.
5. Relate the unknowns in an equation.
6. Solve the equation.
7. Check that the answer makes sense and is consistent with the information given in the problem.
8. Answer the question in the word problem.
Important: Do NOT round any numbers in your calculations until the very final answer!
Helpful Formulas
•
Simple Interest - I = pr (assuming t = 1 )
•
•
•
Uniform Motion - d = rt
Work - W = RT
Percent Mixture - Q = Ar
•
Value Mixture - V = AC
Example – Tara paid one-half of her gameshow winnings to the government for taxes. She invested onethird of her winnings in Jeff’s copy shop at 14% interest and one-sixth of her winnings in Kaiser’s German
Bakery at 12% interest. If she earned a total of $4000 on the investments in one year, then how much
did she win on the game show?
Example – Batman can clean up all of the crime in Gotham City in 8hrs working alone. Robin can do the
same job alone in 12hr. If Robin starts crime fighting at 8am and Batman joins him at 10am, then at
what time will they have all of the crime cleaned up?
Break
You may now use your iPad.
Interactive Lecture (20 minutes)
Which problem should we do next?
(Submit your selection using the ‘Multiple Choice’ box on the Student Response Webpage.)
A. Peter plans to fence off a square
feed lot and then cross-fence to divide
the feed lot into four smaller square
feed lots. If he uses 480ft of fencing,
then how much area will be fenced in?
B. How much water must Poison Ivy
add to a 4-liter solution that contains
5% extract of baneberry to get a
solution that contains 3% extract of
baneberry?
C. The owner of a health-food store
D. Solve for b1 .
sells dried apples for $1.20 per quarter1
pound and dried apricots for $1.80 per =
A
h ( b1 + b2 )
2
quarter-pound. How many pounds of
each must he mix together to get 20
pounds of a mixture that sells for $1.68
per quarter-pound?
Consider the following problem and partial answer.
(Submit your answer using the ‘Multiple Choice’ box on the Student Response Webpage.)
Junior drove his rig on I-10 from San Antonio to El Paso. At the halfway point he noticed that he had
been averaging 80mph, while his company requires his average speed to be 60mph. What must be his
speed for the last half of the trip so that he will average 60mph for the entire trip?
Answer:
The formula that applies to this scenario is d = rt . This applies to the first half of the trip, the second
half of the trip, and to the trip as a whole.
First half
Second half
Whole trip
r t d
Next, fill in what we know:
r
t d
80mph
First half
Second half
Whole trip 60mph
We don’t know much. This problem is strange in that we are not told anything about the time or the
distance. Usually, we’ll have to know something about at least one of these two things in order to get a
solution. In this case, we will have to do without.
We know that the distance of the first half of the trip is the same as the distance of the second half of
the trip, which we can call d. We also know that the whole trip has a distance of 2d. (Why can’t we say
the same thing with the time?)
r
t d
First half
80mph
d
Second half
d
Whole trip 60mph
2d
We’ll let x represent the speed of the second half of the trip. We’ll thus have to fill in the time column
using the formula d = rt .
r
t d
First half
80mph
d
Second half x
d
Whole trip 60mph
2d
We’ll thus have to fill in the time column using the formula d = rt .
r
t
First half
80mph ?
Second half x
Whole trip 60mph
d
d
d
2d
What should the question mark equal?
A.
B.
d
80
80
d
C.
D.
80d
80 − d
Work on the following problem, either alone or with your classmates.
(Submit your answer using the ‘Short Answer’ box on the Student Response Webpage.)
How long does it take an SR-71 Blackbird, one of the fastest U.S. jets, to make a surveillance run of 5570
miles if it travels at an average speed of Mach 3 (2228mph)?
End