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Math Help - geometric series word problem

  1. #1
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    geometric series word problem

    A competitor is running in a 25 km race. For the first 15 km, she runs at a steady rate of 12 km h-1. After completing 15 km, she slows down and it is now observed that she takes 20% longer to complete each kilometre than she took to complete the previous kilometre.

    (a) Find the time, in hours and minutes, the competitor takes to complete the first 16 km of the race.

    The time taken to complete the rth kilometre is  U_{r} hours.

    (b) Show that, for 16 \leq r \leq 25

      u_{r} = \frac{1}{12}(1.2)^{r-15.}

    (c) Using the answer to (b), or otherwise, find the time, to the nearest minute, that she takes to complete the race.
    for part 'a' I got

    time = distance/speed

    15/12 = 1.25 hours
    so 1 hour and 15 minutes for the first 15km?

    after that it takes 20% longer, do I just multiply 1km by 1.20 ?

    dont really know how to do part 'b' and 'c'.
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by Tweety View Post
    for part 'a' I got

    time = distance/speed

    15/12 = 1.25 hours
    so 1 hour and 15 minutes for the first 15km?

    after that it takes 20% longer, do I just multiply 1km by 1.20 ?

    dont really know how to do part 'b' and 'c'.
    The time it took for the 15th kilometer, assuming a steady pace, is 1.25 hours divided 15 , which gives 1/12 hours.

    The 16th kilometer will take 1.20 * (1/12)
    The 17th kilometer will take 1.20 * 1.20 * (1/12)
    ...

    Can you generalize for the r-th kilometer above 15?

    For part c) you can either add the finite list up or the formula for the sum of a geometric sequence.

    Good luck!!
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