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Math Help - Geometric Series Question

  1. #1
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    Post Geometric Series Question

    Bryan is building his own home and is trying to determine whether he should install a conventional heating system or a geothermal heating system in his home.
    Geothermal systems utilize the heat of the earth to provide energy to heat the home. The geothermal system costs $50 000 to install but annual operating costs would be $1200.

    A conventional heating system, which utilizes electricity and natural gas to produce heat for the home, costs only $10 000 initially but annual operational costs would be $3600.

    The operational cost of both systems increases at a rate of about 3% per year.

    Simplify the equation you create as much as possible. The resulting equation should be an exponential equation in the form mx = n. To solve this type of equation, use the trial and error method.

    Determine an equation that could be used to determine the # of years when the total cost is equivalent:

    I'm really bad at math and I'm not even sure where to start...

    I figured out this though:

    Sn = 1200 (0.03^n - 1) / 0.03 - 1

    Formula for the sum of the annual operating costs for n years for a geothermal system.


    Sn = 3600 (0.03^n - 1) / 0.03 - 1

    Formula for the sum of the annual operating costs for n years for a conventional system.


    Any help would be appreciated!
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  2. #2
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    Re: Geometric Series Question

    Hello, Mathnood768!

    Bryan is building his own home and is trying to determine whether he should install
    a conventional heating system or a geothermal heating system in his home.

    Geothermal systems, which utilize the heat of the earth to provide energy to heat the home,
    costs $50 000 to install and annual operating costs would be $1200.

    A conventional heating system, which utilizes electricity and natural gas to heat the home,
    costs only $10 000 initially and annual operating costs would be $3600.

    The operating cost of both systems increases at a rate of about 3% per year.

    Simplify the equation you create as much as possible.
    The resulting equation should be an exponential equation in the form: m^x  = n
    To solve this type of equation, use the trial and error method.

    Determine an equation that could be used to determine the # of years when the total cost is equivalent.

    For n years, the geothermal system would cost a total of:
    . . S_G \;=\;50,\!000 + 1200\left(\frac{1.03^n-1}{0.03}\right\text{ dollars}

    For n years, the conventional system would cost a total of:
    . . S_C \;=\;10,\!000 + 3600\left(\frac{1.03^n-1}{0.03}\right)\text{ dollars}


    If the total costs are equal, we have:

    . . 10,\!000 + 3600\left(\frac{1.03^n-1}{0.03}\right) \;=\;50,\!000 + 1200\left(\frac{1.03^n-1}{0.03}\right)

    . . 2400\left(\frac{1.03^n-1}{0.03}\right) \;=\;40,\!000 \quad\Rightarrow\quad 1.03^n-1 \:=\:0.5 \quad\Rightarrow\quad 1.03^n \:=\:1.5


    Take logs: . \ln(1.03^n) \:=\:\ln(1.5) \quad\Rightarrow\quad n\ln(1.03) \,=\,\ln(1.5)

    . . . . . . . . . n \:=\:\frac{\ln(1.5)}{\ln(1.03)} \:=\:13.71723742


    The total costs are equal in about 13.7 years.
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