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Question

Some of your friends running speeds are between 400 to 500 meters per minute. If they take part in a 4000m race, calculate the shortest time by writing an inequality to model this situation.

Answer 1

This is a D-I-R-T (Distance-Is-Rate-times-Time) problem.  We are given the range of rates; the problem asks for an expression of the shortest time (but we can calculate the range of times).

 

Given  D = R*T          (D-I-R-T)

        R = D/T

 

The race is 4000 m.

 

Given:

      400 meters/min ≤ R ≤ 500 meters/min    (you should determine whether "between" means < or ≤)

     400 meters/min ≤ D/T ≤ 500 meters/min        (substitute D/T for R)

       400 meters/min ≤ (4000m)/T ≤ 500 meters/min     (substitute 4000m for D)

      1/400 min/meter ≥ T/(4000m) ≥ 1/500 meters/min  
                  (note: invert reverses sense of inequality)

       (4000m)(1 min/ 400m) ≥ T ≥ (4000m)(1 meter / 500 min)

                   10 min  ≥ T ≥ 8 min

 

We would usually write this as:

                  8 min ≤ T ≤ 10 min

 

The range of times is between 8 and 10 minutes.  The minimum time is 8 minutes.  T ≥ 8 minutes.

Answer 2

Between suggests not equal to, so my approach to this answer is that neither 400m/min nor 500m/min are included.

 

400 meters/minute < running pace(r) < 500 meters/minute

 

400 < r < 500

 

Therefore, the fastest running speed is less than 500m/min so for a 4000m race, (4000m / 500m/min), the fastest finish time(t) would be greater than 8 minutes! The slowest running speed is greater than 400m/min and so the slowest finish time would be 10 minutes.

 

Shortest run time > 8 minutes

Longest run time < 10 minutes

 

10 minutes > t > 8 minutes

 

If the term between means inclusive of 400 and 500, then:

 

400m/m ≤ r ≤ 500m/m

 

and

 

10 minutes ≥ t ≥ 8 minutes

 

 

I hope this helps!

 

Neal