Answer:
Explanation:
Objective: Solve distance problems by creating and solving a linear
equation.
An application of linear equations can be found in distance problems. When
solving distance problems we will use the relationship rt = d or rate (speed) times
time equals distance. For example, if a person were to travel 30 mph for 4 hours.
To find the total distance we would multiply rate times time or (30)(4) = 120.
This person travel a distance of 120 miles. The problems we will be solving here
will be a few more steps than described above. So to keep the information in the
problem organized we will use a table. An example of the basic structure of the
table is blow:
Rate Time Distance
Person 1
Person 2
Table 1. Structure of Distance Problem
The third column, distance, will always be filled in by multiplying the rate and
time columns together. If we are given a total distance of both persons or trips we
will put this information below the distance column. We will now use this table to
set up and solve the following example
1
Example 1.
Two joggers start from opposite ends of an 8 mile course running towards each
other. One jogger is running at a rate of 4 mph, and the other is running at a
rate of 6 mph. After how long will the joggers meet?
Rate Time Distance
Jogger 1
Jogger 2
The basic table for the joggers, one and two
Rate Time Distance
Jogger 1 4
Jogger 2 6
We are given the rates for each jogger.
These are added to the table
Rate Time Distance
Jogger 1 4 t
Jogger 2 6 t
We only know they both start and end at the
same time. We use the variable tfor both times
Rate Time Distance
Jogger 1 4 t 4t
Jogger 2 6 t 6t
The distance column is filled in by multiplying
rate by time
8 We have total distance, 8 miles, under distance
4t + 6t = 8 The distance column gives equation by adding
10t = 8 Combine like terms, 4t + 6t
10 10 Divide both sides by 10
t =
4
5
Our solution fort, 4
5
hour(48 minutes)
As the example illustrates, once the table is filled in, the equation to solve is very
easy to find. This same process can be seen in the following example
Example 2.
Bob and Fred start from the same point and walk in opposite directions. Bob
walks 2 miles per hour faster than Fred. After 3 hours they are 30 miles apart.
How fast did each walk?
Rate Time Distance
Bob 3
Fred 3
The basic table with given times filled in
Both traveled 3 hours
2
Rate Time Distance
Bob r + 2 3
Fred r 3
Bob walks 2 mph faster than Fred
We know nothing about Fred,so use r for his rate
Bob is r + 2,showing 2 mph faster
Rate Time Distance
Bob r + 2 3 3r + 6
Fred r 3 3r
Distance column is filled in by multiplying rate by
Time.Be sure to distribute the 3(r +2)for Bob.
30 Total distance is put under distance
3r + 6+ 3r = 30 The distance columns is our equation, by adding
6r +6 = 30 Combine like terms 3r + 3r
− 6 − 6 Subtract 6 from both sides
6r = 24 The variable is multiplied by 6
6 6 Divide both sides by 6
r =4 Our solution for r
Rate
Bob 4+2 =6
Fred 4
To answer the question completely we plug 4 in for
r in the table.Bob traveled 6 miles per hour and
Fred traveled 4 mph
Some problems will require us to do a bit of work before we can just fill in the
cells. One example of this is if we are given a total time, rather than the individual times like we had in the previous example. If we are given total time we
will write this above the time column, use t for the first person’s time, and make
a subtraction problem, Total − t, for the second person’s time. This is shown in
the next example
Example 3.
Two campers left their campsite by canoe and paddled downstream at an average
speed of 12 mph. They turned around and paddled back upstream at an average
rate of 4 mph. The total trip took 1 hour. After how much time did the campers
turn around downstream?
Rate Time Distance
Down 12
Up 4
Basic table for down and upstream
Given rates are filled in
1 Total time is put above time column
Rate Time Distance
Down 12 t
Up 4 1 − t
As we have the total time, in the first time we have
t,the second time becomes the subtraction,
total − t
3
Rate Time Distance
Down 12 t 12t
Up 4 1 − t 4 − 4t
=
Distance column is found by multiplying rate
by time.Be sure to distribute 4(1 − t)for
upstream. As they cover the same distance,
= is put after the down distance
12t = 4 − 4t With equal sign, distance colum is equation
+ 4t + 4t Add4tto both sides so variableis onlyon one side
16t =4 Variable is multiplied by 16
16 16 Divide both sides by 16
t =
1
4
Our solution,turn around after 1
4
hr(15 min )
Another type of a distance problem where we do some work is when one person
catches up with another. Here a slower person has a head start and the faster
person is trying to catch up with him or her and we want to know how long it
will take the fast person to do this. Our startegy for this problem will be to use t
for the faster person’s time, and add amount of time the head start was to get the
slower person’s time. This is shown in the next example.