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13 April, 16:01

The volume of one cloud, in cubic kilometers, grows according to the expression 4^3x - 1, where x is time in hours. Another cloud grows according to the expression 8x. After how many hours will the two clouds have the same volume? What is their volume, in cubic kilometers?

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  1. 13 April, 16:28
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    x = ⅔ hr

    V = 4 km³

    Step-by-step explanation:

    The volume of the first cloud is given as:

    V1 = 4^ (3x - 1)

    The volume of the second cloud is given as:

    V2 = 8^x

    When the two clouds have the same volume, V1 = V2:

    => 4^ (3x - 1) = 8^x

    We can rewrite this as:

    2^[2 (3x - 1) ] = 2^[3 (x) ]

    => 2^ (6x - 2) = 2^ (3x)

    According to the law of indices, we can equate the powers because their bases are equal (2 is the base).

    => 6x - 2 = 3x

    6x - 3x = 2

    3x = 2

    x = ⅔ hr

    Hence, after 2/3 hr, their volumes will be equal.

    To find the volume after 2/3 hr, we can use the equation of either cloud but let us use the equation for the second cloud for simplicity sake:

    V2 = 8^ (⅔)

    V2 = 4 km³ or 4 cubic kilometers
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