Village bought

A village bought in the initial year for winter maintenance needs a tractor at a purchase price of 70 000 €. Its value decreases exponentially and after two years has a value of 49 000 €. The maintenance costs grow exponentially, in the first year they are 1 200 € and in the second year they are 2 500 €.

A Find the optimal lifetime of the tractor A:
B - calculation of the average costs for A+1
C - calculation of the residual value of the tractor in the year of optimal lifetime
D - calculation of the maintenance costs in the third year

Many thanks

Final Answer:

A =  4.7282
C =  16807 Eur
D =  5208.3333 Eur

Step-by-step explanation:

$$\begin{gathered} a_1=70000\text{Eur} \\ {a}_2=49000\text{Eur} \\ q=a_2/a_1=49000/70000=\frac{7}{10}=0.7 \\ {n}_1=1200\text{Eur} \\ {n}_2=2500\text{Eur} \\ r=n_2/n_1=2500/1200=\frac{25}{12}=2\frac{1}{12}\doteq 2.0833 \\ {a}_1\cdot q^{A-1}=n_1\cdot r^{A-1} \\ \ln a_1+(A-1)\cdot \ln q=\ln n_1+(A-1)\cdot \ln r \\ A=1+\frac{\ln (n_1)-\ln (a_1)}{\ln q-\ln r}=1+\frac{\ln 1200-\ln 70000}{\ln 0.7-\ln 2.0833}=4.7282 \end{gathered}$$

$D=n_1\cdot r^2=1200\cdot {\frac{25}{12}}^2=5208.3333\text{Eur}$

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