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30 April, 16:27

A marathon is 26.2 miles. What is the least number of times Miguel must run for his total distance run during training to exceed the distance of a marathon?

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  1. 30 April, 16:36
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    Missing part of the question

    Miguel has started training for a race. The first time he trains, he runs 0.5 mile. Each subsequent time he trains, he runs 0.2 mile farther than he did the previous time.

    What is the arithmetic series that represents the total distance Miguel has run after he has trained n times?

    Answer:

    The least number of times Miguel must run for his total distance run during training to exceed the distance of a marathon is 17.3 miles

    Step-by-step explanation:.

    Given parameters

    Miguel first run = 0.5 mile

    Subsequent run = 0.2 mile

    This question is an arithmetic progression.

    We'll make use of arithmetic progression formula to solve this

    Formula:.

    Tn = a + (n - 1) d

    Where a = first term

    n = number of terms

    d = common difference

    In this case

    a = first run = 0.5 mile

    d = subsequent run = 0.2 mile

    So, Tn = a + (n - 1) d become

    Tn = 0.5 + (n - 1) 0.2

    Tn = 0.5 + 0.2n - 0.2

    Tn = 0.5 - 0.2 + 0.2n

    Tn = 0.3 + 0.2n

    The arithmetic series of an arithmetic progression is calculated using

    Sn = ½ (a + Tn) * n

    By substituton, we have

    Sn = ½ (0.5 + 0.3 + 0.2n) * n

    Sn = ½ (0.8 + 0.2n) * n

    Sn = 0.4n + 0.1n²

    b.

    Since the race is 26.2 miles then the least number of times is given as

    Sn ≥ 26

    0.4n + 0.1n² ≥ 26.2

    0.1n² + 0.4n - 26.2 ≥ 0

    Using quadratic formula

    n = (-b ± √ (b² - 4ac)) / 2a

    Where b = 0.4 a = 0.1 and C = - 26.2

    So,

    n = - 0.4 ± √ (0.4² - 4 * 0.1 *,26.2) / 2 * 0.1

    n = (-0.4 ± √10.64) / 0.2

    n = (0.4 ± 3.26) / 0.2

    n = (0.4 + 3.26) / 0.2 or (0.4 - 3.26) / 0.2

    n = 3.46/0.2 or - 2.86/0.2

    n = 17.3 or - 14.3

    Since n can't be negative

    n = 17.3 miles

    The least number of times Miguel must run for his total distance run during training to exceed the distance of a marathon is 17.3 miles
  2. 30 April, 19:54
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    15 times
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