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C2 . . . . . Everyone Race to this Question! ! Geometric Series.

A runner is running in a 25km race. For the 1st 15 km, she runs at a steady rate of 12km/h

After completing 15km, she slows down and is now observed that she takes 20% longer to complete each km than she took to complete the previous km.

Find the time in hrs and mins the runner takes to complete the 1st 16km of the race.

I guarantee rep to the 1st ...plz explain step by step

Reply 1

generalebriety

I should point out that you only have 88 posts and your rep is consequently worth nothing. But it's a nice thought. :wink:

How long does it take her to do the first 15km? She then takes 20% longer, so what's her speed on the 16th kilometre? How long does she take to do that?

Reply 2

gorilla_baby

(a) Using time =distance
speed
=15
12
= 1.25 hours = 1 hour 15 mins.
The competitor takes 1 hour 15 mins for the first 15 km.
Time for each km is 1hour15mins
15
=75
15
=5 mins
Time for the 16th km is 5×1.2=6 mins
Total time for first 16 km is 1 hour 15 mins + 6 mins = 1 hour 21 mins

(b) Time for the 17th km is 5×1.2×1.2=5×1.22 mins
Time for the 18th km is 5×1.23 mins
Time for the rth km is 5×(1.2)r−15 mins =5×(1.2)r−15
60
hours
So ur=1
12
(1.2)r−15

(c) Consider the 16th to the 25th kilometre.
Total time for this distance
=5×1.2+5×1.22+5×1.23+ … + 5×1.210
=5× 

A geometric series with a = 1.2, r = 1.2 and n = 10.
=5×1.2(1.210−1)
1.2−1

= 155.75 mins
= 156 mins (to the nearest minute)
Total time for the race
= time for 1st 15 km + time for last 10 km
=75+156
= 231 mins
= 3 hours 51 mins
:biggrin:

Reply 3

bubble_gum_pop

I forgot to add.... the answer is 1 hr 21 mins

i got 1 hr 19 mins.... :s-smilie:

Reply 4

barry_4_england

3km will take her 15 mins, so 1hr+15 mins for the 15km.. 1 km will take her 5 minutes normally, but the last 1(16th) will take her 6 mins. Add em up and you get 1 hr 21.

Reply 5

bubble_gum_pop

thx

can you prove that for the nth km , which is in Ur hours

for 16 ≤ r ≤ 25 , Ur = 1/12 (1.2) ^r-15

please, i can't get it :frown:

EDIT: the details for the question are at the top of this thread

Reply 6

generalebriety

bubble_gum_pop

thx

can you prove that for the nth km , which is in Ur hours

for 16 ≤ r ≤ 25 , Ur = 1/12 (1.2) ^r-15

please, i can't get it :frown:

EDIT: the details for the question are at the top of this thread

I'm using t for time, I'm not entirely sure what you're using, but I'm sure you'll understand.

t₁₅ = distance / speed = 1/12.
t₁₆ = t₁₅ * 1.2
t₁₇ = t₁₆ * 1.2 = t₁₅ * (1.2)^2
t₁₈ = t₁₇ * 1.2 = t₁₆ * (1.2)^2 = t₁₅ * (1.2)^3...

Reply 7

phantom1813

what gorilla baby has done is absolutely correct! The answer to part (c) in the heinemann c2 book is stated as 3hrs 25 mins which is WRONG. It is an error which caused me a lot of confusion as i had gotten 3 hrs 51mins which is indeed the correct answer.