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A bacteria doubles every hour. If there are 23 bacteria at 8:00 am, how many will there be at 8:00 pm. Express using powers of 2 and find the actual number.

Mathematics

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Answer

2152^{15}

32768

Step-by-step solution

4 steps

Here's the solution to the problem:

  1. Determine the initial number of bacteria and the time duration.

    The initial number of bacteria at 8:00 AM is given as 232^3.
    The time duration from 8:00 AM to 8:00 PM is 12 hours.

  2. Calculate the number of doublings.

    Since the bacteria double every hour, over 12 hours, the bacteria will double 12 times. This means the initial number will be multiplied by 2122^{12}.

  3. Express the total number of bacteria using powers of 2.

    To find the total number of bacteria at 8:00 PM, multiply the initial number by the doubling factor:

    Total bacteria=Initial bacteria×2number of hours\text{Total bacteria} = \text{Initial bacteria} \times 2^{\text{number of hours}} Total bacteria=23×212\text{Total bacteria} = 2^3 \times 2^{12}

    Using the exponent rule am×an=am+na^m \times a^n = a^{m+n}:

    Total bacteria=23+12\text{Total bacteria} = 2^{3+12} Total bacteria=215\text{Total bacteria} = 2^{15}

  4. Calculate the actual number of bacteria.

    215=327682^{15} = 32768

    The number of bacteria at 8:00 PM, expressed using powers of 2, is 215\boxed{2^{15}}.
    The actual number of bacteria is 32768\boxed{32768}.

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