AP lap times against GP lap times
A marathon runner and a sprinter run laps on a 400m track. The sprinter takes 56 seconds for lap 1, then 6 seconds more each subsequent lap. The marathon runner takes 1 minute for lap 1, then 4% more than the previous lap.
(i) Show the sprinter’s time for laps is . marks
(ii) Find the marathon runner’s time for laps in terms of . marks
(iii) How many complete laps does it take for the marathon runner to first exceed 40 minutes total? marks
They decide to do a 12km race with the same starting point.
(iv) Determine who finishes first. marks
(v) Determine which lap the winner is on when first overtaking the other runner. marks
[2023/JPJC/Prelim/I/8]
Solution
Sprinter’s cumulative time
Show that the sprinter takes seconds for laps.
Hint
Do not confuse the nth lap time with the total for n laps.
Answer · (i)
seconds.
Sum n lap times
Lap takes seconds. The race time is the sum of these lap times.
Marathon runner’s cumulative time
Find the marathon runner’s total time for laps.
Hint
A 4% increase means ratio 1.04, not 0.04.
Answer · (ii)
seconds.
Sum the geometric lap times
Because r>1, use the equivalent positive-denominator version of the finite sum formula.
First complete-lap total above 40 minutes
Find the first completed lap count for which the marathon runner’s elapsed time exceeds 40 minutes.
Hint
Convert 40 minutes to seconds before substituting.
Answer · (iii)
complete laps.
Solve the strict inequality
The logarithm denominator is positive this time, so its direction is unchanged.
Check the adjacent counts
The cumulative time increases, so these neighbouring counts establish minimality.
Who finishes 12 km first?
Determine the winner of the 12 km race.
Hint
First translate the race distance into a lap count.
Answer · (iv)
Marathon runner: seconds against seconds.
Count the laps
Use common distance for both runners.
Compare totals
Both formulas return seconds.
Lap of the first lead change
Find the lap of the winner’s first overtake.
Hint
Compare both cumulative times at n=3 and n=4.
Endpoint totals do not supply an exact within-lap overtaking time. Retain the source’s lap 4 interpretation with that qualification.
Answer · (v)
Lap .
Endpoint totals establish the lead change; a precise within-lap crossing requires a speed model.
Check before the change
After three laps, the sprinter has the lower cumulative time.
Check after the change
After four laps, the marathon runner has the lower cumulative time.
Alternative: compare the cumulative-time graphs
The first positive intersection of these extended total-time formulas is near . Check the integer rows on either side: the order changes between laps 3 and 4. A fractional lap index in this graph does not specify the actual overtaking instant; that still requires a model for speed within each lap.
State what the model establishes
The source reports lap 4. A precise within-lap crossing requires a speed model not specified by these lap totals.
Explore the idea
Compare elapsed times over the same distance
Cumulative times at the same lap boundary identify when the lead changes between completed laps.