Question

The value of a machine. V at the end of years is given by V = C(1-r)^t, where C is the original cost of the machine and r is the rate of depreciation. A machine thatoriginally cost $13,500 is now valued at $10,419. How old is the machine if r= 0.12? Round your answer to two decimal places.AnswerHow to enter your answer (Opens in new window)KeypadKeyboard Shortcuts

Answer 1

The value of a machine at the end of t years with a rate of depreciation, r is given by the function:

`V=C(1-r)^t`

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:

`\begin{gathered} 10419=13500(1-0.12)^t \\ \Rightarrow10419=13500(0.88)^t \end{gathered}`

Solve the resulting equation for t:

`\begin{gathered} \text{swap the equation:} \\ \Rightarrow13500(0.88)^t=10419 \\ Divide\text{ both sides by 13500:} \\ \Rightarrow(13500(0.88)^t)/(13500)=(10419)/(13500) \\ \Rightarrow0.88^t=(3473)/(4500) \end{gathered}`

Take the Logarithm of both sides:

`\begin{gathered} \ln(0.88^t)=\ln((3473)/(4500)) \\ \Rightarrow t\cdot\ln(0.88)=\ln((3473)/(4500)) \\ \Rightarrow t=(\ln((3473)/(4500)))/(\ln(0.88))\approx2.03 \end{gathered}`

Hence, the machine is about 2.03 years.

The machine is about 2.03 years.