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 *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:

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!

Re: Geometric Series Question

Hello, Mathnood768!

Quote:

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: $\displaystyle m^x = n$

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 $\displaystyle n$ years, the geothermal system would cost a total of:

. . $\displaystyle S_G \;=\;50,\!000 + 1200\left(\frac{1.03^n-1}{0.03}\right\text{ dollars}$

For $\displaystyle n$ years, the conventional system would cost a total of:

. . $\displaystyle S_C \;=\;10,\!000 + 3600\left(\frac{1.03^n-1}{0.03}\right)\text{ dollars}$

If the total costs are equal, we have:

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

. . $\displaystyle 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: .$\displaystyle \ln(1.03^n) \:=\:\ln(1.5) \quad\Rightarrow\quad n\ln(1.03) \,=\,\ln(1.5) $

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

The total costs are equal in about 13.7 years.