The value of a machine at the end of t years with a rate of depreciation, r is given by the function:
[tex]V=C(1-r)^t[/tex]
It is required to find the age of a machine that originally costs $13,500, but is now valued at $10,419 and has a rate of depreciation of 0.12.
To do this, substitute C=13500, V=10419, and r=0.12 into the function:
[tex]\begin{gathered} 10419=13500(1-0.12)^t \\ \Rightarrow10419=13500(0.88)^t \end{gathered}[/tex]
Solve the resulting equation for t:
[tex]\begin{gathered} \text{swap the equation:} \\ \Rightarrow13500(0.88)^t=10419 \\ Divide\text{ both sides by 13500:} \\ \Rightarrow\frac{13500(0.88)^t}{13500}=\frac{10419}{13500} \\ \Rightarrow0.88^t=\frac{3473}{4500} \end{gathered}[/tex]
Take the Logarithm of both sides:
[tex]\begin{gathered} \ln(0.88^t)=\ln(\frac{3473}{4500}) \\ \Rightarrow t\cdot\ln(0.88)=\ln(\frac{3473}{4500}) \\ \Rightarrow t=\frac{\ln(\frac{3473}{4500})}{\ln(0.88)}\approx2.03 \end{gathered}[/tex]
Hence, the machine is about 2.03 years.
The machine is about 2.03 years.
