Respuesta :
By applying formula of Present value of annuity we got that after eight years, percentage of the total lifetime cost of the system did the original price make up is 38.25%
What is Present value of annuity?
The current value of future payments from an annuity, given a specified rate of return, or discount rate is known as Present value of annuity.
We know that formula of Present value of annuity is
[tex]\text{PV}=P\left[\frac{1-(1+r)^{-n}}{r}\right][/tex]
Where
P= Payment per period (monthly)
r= Rate of interest per period =10.31 % annually =0.1031/12 monthly
n= Number of periods =4 years =48 months
So Payment per period can be calculated as :
[tex]$\begin{aligned}&\Rightarrow 1874=P\left[\frac{1-\left(1+\frac{0.1031}{12}\right)^{-48}}{\frac{0.1031}{12}}\right] \\\\&\Rightarrow P=\frac{1874}{\left[\frac{\left.1-\left(1+\frac{0.9331}{12}\right)^{-4 .}\right]}{\frac{0.121}{12}}\right]} \\\\&\Rightarrow P=\$ 47.81\end{aligned}$[/tex]
Total payment =47.81[tex]\times[/tex]48=2294.88
Given that over the eight years that Olivia kept the sprinkler system, it used an average of $ 2.11 in water per week.
So total amount [tex]$=2.11 \times 52 \times 8$=\$ 877.76$[/tex]
The percentage of the total lifetime cost of the system did the original price make up can be calculated as
[tex]$\begin{aligned}&=\frac{877.76}{2294.88} \times 100 \% \\\\&=38.25 \%\end{aligned}$[/tex]
By applying formula of Present value of annuity we got that after eight years, percentage of the total lifetime cost of the system did the original price make up is 38.25%
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