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Math Help - Exponential and Logarithmic Equations

  1. #1
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    Question Exponential and Logarithmic Equations

    Good Morning,

    My problem is: Use a calculator with a y^x key or ^ key to solve the following.

    The 1986 explosion at Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The functon f(x)=1000(0.5)^ x over 30 describes the amount, f(x), in kilograms of cesium-137 remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyls atmosphere, the area is considered unsafe for human habitation. Find f(80) and determine if Chernobyl will be safe for human habitation by 2066.
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  2. #2
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    Quote Originally Posted by kbryant05
    Good Morning,

    My problem is: Use a calculator with a y^x key or ^ key to solve the following.

    The 1986 explosion at Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The functon f(x)=1000(0.5)^ x over 30 describes the amount, f(x), in kilograms of cesium-137 remaining in Chernobyl x years after 1986. If even 100 kilograms of cesium-137 remain in Chernobyls atmosphere, the area is considered unsafe for human habitation. Find f(80) and determine if Chernobyl will be safe for human habitation by 2066.
    Hello,

    I hope that you are allowed to use the ln-button too.

    Plug in the value(s) you know into the equation:

    f(x)=1000 \cdot \left( \frac{1}{2} \right)^{\frac{x}{30}

    That means you have to solve the equation for x:

    80=1000 \cdot \left( \frac{1}{2} \right)^{\frac{x}{30}

    The next steps should be:

    0.08=\left( \frac{1}{2} \right)^{\frac{x}{30}

    The exponent x/30 is a logarithm to the base 0.5. Use the base-change-formula (I am not sure, if this is the right expression. But maybe you recognize the name when you have a look at the solution)

    \frac{\ln(0.08)}{\ln(0.5)}=\frac{x}{30}

    So:

    30\cdot\frac{\ln(0.08)}{\ln(0.5)}=x. That means x is round about 110 years. In 2066 Chernobyl will be one of the most dangerous places on earth.

    Greetings

    EB
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  3. #3
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    Reply on problem

    Thanks Earboth. I don't think that answer is very comforting for anyone living in that area. It's actually really frightening.
    Yes we are allowed to use the in button too!!
    Have a great day!
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  4. #4
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by earboth
    That means x is round about 110 years. In 2066 Chernobyl will be one of the most dangerous places on earth.

    Greetings

    EB
    I don't agree. You said that  f(x)=f(80)=80 , which would solve the equation for "how many years does it take to reach 80kg" but the question says it is  f(x)=f(80) so  x=80

    So you would substitute 80 for x instead of the function of x.

     f(x)=1000(0.5)^{\frac{x}{30}}

     f(80)=1000(0.5)^{\frac{80}{30}}

     f(80)=1000(0.5)^{\frac{8}{3}}

     f(80)=1000(0.157490131236859)

     f(80)=157.490131236859

    So, yes it is still lethal.
    Last edited by Quick; June 15th 2006 at 01:01 PM.
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