Lesson 4: Graphing Linear Equations and Inequalities
Problem Solving Assignment solutions
In the year 1923, the men’s world record for the mile was set at 4:10.4
minutes. Roger Bannister broke the 4-minute mile in 1954 with a time of
3:59.4 minutes.
a) Assume that the world record follows a linear pattern. Use the two
data points above to find an equation that relates the year (x) to the
world record time in the mile (y). Write the equation in slopeintercept form. Note: Leave the slope as a fraction. Round the yintercept to the nearest hundredth.
If we represent the time in seconds, the two data points are
(1923,250.4) and (1954,239.4 ) .
The slope is:
m=
y2 − y1
x2 − x1
239.4 − 250.4
1954 − 1923
−11
=
31
=
The y-intercept is:
y = mx + b
−11
250.4 =
(1923) + b
31
250.4 = −682.35 + b
932.75 = b
The equation of the line is:
y =−
11
x + 932.75
31
Algebra 1
© 2009 Duke University Talent Identification Program
Page 1 of 3
Lesson 4: Graphing Linear Equations and Inequalities
Problem Solving Assignment solutions
b) In 1999, Hicham El Guerrouj broke the then-current world record. Use
your equation from part a to predict El Guerrouj’s time.
11
(1999 ) + 932.75
31
= −709.32 + 932.75
y =−
= 223.43
The expected time is 223.43 seconds, which is
3:43.43 minutes.
c) Do a web search to find El Guerrouj’s time. How does this compare to
your prediction in part a?
El Guerrouj’s time was 3:43.13 minutes. The prediction from
part a is only 0.3 seconds off of the actual time.
d) Use your equation to predict the world record for the year 2020.
In the year 2020, we would have:
11
(2020) + 932.75
31
= 215.98
y =−
The expected world record time would be 3:35.98 minutes in
2020.
e) According to this equation, in what year will the 3-minute mile be
broken?
We are looking for the year (x) in which the world record time (y) is
less than 3 minutes (or 180 seconds). To find this, we substitute
180 into the equation and solve for x.
Algebra 1
© 2009 Duke University Talent Identification Program
Page 2 of 3
Lesson 4: Graphing Linear Equations and Inequalities
Problem Solving Assignment solutions
11
x + 932.75
31
11
−752.75 ≥ −
x
31
180 ≥ −
−
31
( −752.75) ≤ x
11
2121.39 ≤ x
x ≥ 2121.39
If this equation is accurate, we would expect the 3-minute mile to
be broken during the year 2121.
f) Calculate the time for the year 2600. Comment on the practicality of
this data point.
We are trying to find the world-record time (x) for the year 2600 (y).
We substitute 2600 for y in our equation and solve for x:
11
(2600) + 932.75
31
= −922.58 + 932.75
y =−
= 10.17
This says that the world record for the mile will be just over 10
seconds in the year 2600. It is highly unlikely that a man will be able
to run a mile in less than 11 seconds.
Algebra 1
© 2009 Duke University Talent Identification Program
Page 3 of 3