Hon 發問於 科學及數學數學 · 4 年前

[F6 MATH] Arithmetic Sequence application?

A submarine dives from the sea level. The submarine dives 500 m in the first minute. The distance dived in each of the following minutes is 25 m less than than that in the previous minute. If the submarine can dive to a maximum of 3000 m from the sea level, after how many minutes will the submarine stop diving?

(Give answer correct to 1 significant figure.)

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    Question:

    A submarine dives from the sea level. The submarine dives 500 m in the first minute. The distance dived in each of the following minutes is 25 m less than that in the previous minute. If the submarine can dive to a maximum of 3000 m from the sea level, after how many minutes will the submarine stop diving?

    (Give answer correct to 1 significant figure.)

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    Solution:

    The submarine dived 500 m in the 1st minute.

    The submarine dived 500 - 25 = 500 - 25×1 m in the 2nd minute.

    The submarine dived 500 - 25 - 25 = 500 - 25×2 m in the 3nd minute.

    ...

    The submarine dived 500 - 25(n - 1) m in the nth minute.

    The total distance in the nth minute

    = 500 + (500 - 25) + (500 - 25×2) + ... + [500 - 25(n - 1)]

    = 500n - 25[1 + 2 + 3 + ... + (n - 1)]

    = 500n - 25(1 + n - 1)(n - 1)/2

    = 500n - 25(n - 1)n/2

    ∵ The submarine can dive to a maximum of 3000 m from the sea level.

    n > 0 and 500n - 25(n - 1)n/2 ≤ 3000

    n > 0 and 1000n - (25n² - 25n) ≤ 6000

    n > 0 and 25n² - 1025n + 6000 ≥ 0

    n > 0 and n² - 41n + 240 ≥ 0

    n > 0 and {n - [41 - √(41² - 4×240)]/2}{n - [41 + √(41² - 4×240)]/2} ≥ 0

    n > 0 and [n - (41 - √721)/2][n - (41 + √721)/2] ≥ 0

    0 < n ≤ (41 - √721)/2 or n ≥ (41 + √721)/2 ...... ( rejected )

    ∴ The submarine will stop diving after 7 min.

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    (41 - √721)/2 ≈ 7.07

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