Transcript Motion Word Problems - TCC: Tidewater Community College

Word Problems - Motion
By
Joe Joyner
Math 04
Intermediate
Algebra
Link to Practice Problems
Introduction
In this module, you’ll continue to develop and work
with mathematical models.
When solving practical application problems,
you try to find a mathematical model for the
problem.
A mathematical model does not necessarily have to
be complicated. It can be relatively simple. This is
usually the case when only one or two variables are
required to build a linear model. Let’s begin.
Rate, Time, and Distance Problems
If an object such as an automobile or an
airplane travels at a constant, or uniform,
rate of speed, “r” ,
then the distance traveled by the object,
“d”, during a period of time, “t”
Rate, Time, and Distance Problems
is given by the “distance, rate, time”
formula: d = rt.
Rate, Time, and Distance Problems
Example 1
You ride your bike for 7 hours. If you travel
36.75 miles, what is your average speed?
Rate, Time, and Distance Problems
Example 1
The quantities in this problem are:
• distance (constant at 36.75 miles),
• time (constant at 7 hours),
• and rate, or speed (unknown
variable).
Rate, Time, and Distance Problems
Example 1
You can use a spreadsheet (Excel, for
example) to build a model for this problem.
Rate, Time, and Distance Problems
Example 1
Explore
Biker Problem
To access the
spreadsheet,
click the word
Explore.
Rate * Time = Distance
d = rt
Rate (mph) Time (h)
Biker
Distance (m)
7
0
Then explore with the rate to see if you can
solve the problem.
Rate, Time, and Distance Problems
Example 1
Represent the variable rate with r .
You can use the distance, rate,
time formula.
d = rt
Rate, Time, and Distance Problems
Example 1
But since you know the distance and time, and
wish to solve for rate, it would be helpful to
solve the equation for r first.
d
r
t
Rate, Time, and Distance Problems
Example 1
d
r
t
is our mathematical model.
Some mathematical models can be easy!
Rate, Time, and Distance Problems
Example 1
Now we can solve for the rate, r , by dividing
the distance by the time.
d 36.75
r 
 5.25 miles per hour
t
7
Rate, Time, and Distance Problems
When you read a word problem that
involves rate, time, and distance, note
whether the problem situation involves
• motion in the same direction;
• motion in opposite directions;
• a round trip.
Rate, Time, and Distance Problems
Example 2
Dan and Emily are truck drivers. Dan, averaging
55 miles per hour (mph), begins a 280-mile trip
from their company’s Norfolk warehouse to
Charlotte, NC at 7 AM.
Emily sets out from the Charlotte warehouse at
8 AM on the same day as Dan and travels at
45 mph in the opposite direction as the route taken
by Dan.
Rate, Time, and Distance Problems
Example 2
How many hours will Emily have been driving
when she and Dan pass each other?
How will you start to set up a model for solving
this problem?
Rate, Time, and Distance Problems
Example 2
What is the variable that you must solve for?
time
Is the length of time traveled the same for
Dan and Emily when they pass each other?
No.
Rate, Time, and Distance Problems
Example 2
Why is the time different for the two drivers?
• Dan started at 7 AM and
• Emily started at 8 AM.
• Dan averaged 55 mph and
• Emily averaged 45 mph.
Rate, Time, and Distance Problems
Example 2
Let t represent the amount of time that Emily
travels until the trucks pass each other.
In terms of t , how long will Dan have been
on the road when the trucks pass each other?
One hour longer or ...
t+1
Rate, Time, and Distance Problems
Example 2
You can use a spreadsheet to build a model for
this problem too.
Rate, Time, and Distance Problems
Example 2
Explore
Truck Driver Problem
Rate * Time = Distance
d = rt
To access the
spreadsheet,
click the word
Explore.
Rate (mph) Time (h) Distance (m)
Dan
55
1
55
Emily
45
0
0
Total Distance
Then explore with Emily’s time to see if you
can solve the problem.
55
Rate, Time, and Distance Problems
Example 2
The mathematical model for this problem is:
Dan’s Distance + Emily’s Distance = 280 miles
Dan’s rate*Dan’s time + Emily’s rate*Emily’s time = 280
55(t+1) + 45t = 280
Rate, Time, and Distance Problems
Example 2
55(t+1) + 45t = 280
55t+55 + 45t = 280
100t + 55 = 280
100t = 225
t = 2.25 hours
Rate, Time, and Distance Problems
Example 3
Jason and LeRoy are entered in a 26-mile marathon
race. Jason’s average pace is 6 miles per hour
(mph) and LeRoy’s average pace is 8 mph. Both
runners start at the same time.
How far from the finish line will Jason be when
LeRoy crosses the finish line?
Rate, Time, and Distance Problems
Example 3
What are the known constants?
• Jason’s rate of 6 mph
• LeRoy’s rate of 8 mph
• Race distance of 26 miles
Rate, Time, and Distance Problems
Example 3
What are the unknowns?
• The amount of time it takes LeRoy to
finish the race
• The distance Jason has to run when
LeRoy finishes
Rate, Time, and Distance Problems
Example 3
Let LeRoy’s time be t .
What is the distance, rate, time,
model for Leroy in this problem?
8t = 26
What is the solution for t ?
t = 3.25 hours
Rate, Time, and Distance Problems
Example 3
At the time that LeRoy crosses the finish
line, Jason has run for the same amount of
time, t .
What is the model for
how far Jason is from the
finish line at that time?
d = 26 - 6(3.25)
d = 6.5 miles
Rate, Time, and Distance Problems
Do you think you’ve got the concept of
solving motion (rate, time distance)
problems?
Look at the next slide.
If you want to try the interactive web site
that the slide came from, click on the word
Explore to go there.
Explore
Rate, Time, and Distance Problems
Hopefully, you are now ready to practice motion
problems for yourself. When you click the Go To
Practice Problems link below, your web browser
will open the practice problem set.
Go To Practice Problems