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Math Help - Geometric Series word problem - Bouncing Ball

  1. #1
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    Geometric Series word problem - Bouncing Ball

    A ball is dropped from a height of 10 m. Each time it strikes the ground it bounces vertically to a height that is 3/4 of the preceding height. Find the total distance the ball will travel if it is assumed to bounce infinitely often.

    Answer: 70 m

    I cannot for the life of my figure this out.

    I tried the obvious way: making the geometric series (10)* (\frac{3}{4})^k and then found the convergence as k --> infinity, and got 40.

    Next, because the ball bounces twice, I thought perhaps I needed a 2. I made the following geometric series,
    (2)(10)*\frac{3}{4}, used the convergence formula, and got 80.

    I believe my second attempt makes more sense, but it doesn't equal 70 m.

    So, what am I doing wrong?

    Thanks!
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  2. #2
    Junior Member
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    You have got it Ė except that you donít multiply the first distance (the initial drop of 10\ \mathrm m) by 2. The distance is therefore

    10+2(10)\left(\frac34\right)+2(10)\left(\frac34\ri  ght)^2+2(10)\left(\frac34\right)^3+\cdots
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  3. #3
    MHF Contributor
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    Quote Originally Posted by Truthbetold View Post
    A ball is dropped from a height of 10 m. Each time it strikes the ground it bounces vertically to a height that is 3/4 of the preceding height. Find the total distance the ball will travel if it is assumed to bounce infinitely often.

    Answer: 70 m

    I cannot for the life of my figure this out.

    I tried the obvious way: making the geometric series (10)* (\frac{3}{4})^k and then found the convergence as k --> infinity, and got 40.

    Next, because the ball bounces twice, I thought perhaps I needed a 2. I made the following geometric series,
    (2)(10)*\frac{3}{4}, used the convergence formula, and got 80.

    I believe my second attempt makes more sense, but it doesn't equal 70 m.

    So, what am I doing wrong?

    Thanks!
    HI

    You have the right idea .

    10 + 2(7.5)+2(5.625)+...

    =10+2[7.5+5.625+...]

    =10+2[7.5/(1-0.75)]

    =70
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