Math 20-1 help please!?

2013-02-14 5:32 pm

Sorry If this is long...
A geothermal system costs $50 000 to install but has an annual operating cost of $1200. A conventional heating system costs $10 000 to install and has an annual operating cost of $3600. **The operational costs of each system increases by 3% per year**

Geometric sequence is: Sn= initial cost+ t1[(r^n)-1]/r-1
Geothermal Sn= 50 000+ 1200[(1.03^n)-1]/1.03-1
Conventional Sn = 10 000+ 3600[(1.03^n)-1]/1.03-1

Question: Determine an equation that could be solved to determine the number of years when the total cost of each system is equivalent.

回答 (1)

An ESL Learner

2013-02-14 6:07 pm

✔ 最佳答案

If the total cost of each system is equivalent, then
50000 + 1200(1.03^n - 1)/(1.03 - 1) = 10000 + 3600(1.03^n - 1)/(1.03 - 1).

Solve for n.
50000 + 1200(1.03^n - 1)/(1.03 - 1) = 10000 + 3600(1.03^n - 1)/(1.03 - 1)
(3600 - 1200)(1.03^n - 1)/0.03 = 50000 - 10000
1.03^n - 1 = 40000/2400 * 0.03
1.03^n = 1 + 0.5
ln(1.03^n) = ln1.5
n * ln1.03 = ln1.5
n = ln1.5/ln1.03
n = 13.717237417

Therefore, the total cost of each system is equivalent, during the 2nd half of the 13th year and before the 14th year.

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