Question

If we take into account its depreciation, the value of a certain machine after t years of use is V (t) = 20,000e ^ (- 0.4t) dollars. a) What is the depreciation rate after 5 years of use? b) What is the relative rate of change in the value of the machine after t years of use.

Answer 1

`\begin{gathered} \text{Given} \\ V(t)=20000e^(-0.4t) \end{gathered}`

a) What is the depreciation rate after 5 years of use?

First, get the derivative of the function V(t), and then substitute t = 5.

`\begin{gathered} V(t)=20000e^(-0.4t) \\ V^(\prime)(t)=20000\cdot-0.4\cdot e^(-0.4t) \\ V^(\prime)(t)=-8000e^(-0.4t) \end{gathered}`
`\begin{gathered} \text{If }t=5,\text{ then }V^(\prime)(t)\text{ is} \\ V^(\prime)(t)=-8000e^(-0.4t) \\ V^(\prime)(5)=-8000e^(-0.4(5)) \\ V^(\prime)(5)=-8000e^(-2) \\ V^(\prime)(5)=-1082.682266 \\ \text{Round off to two decimal place} \\ V^(\prime)(5)=-1082.68 \end{gathered}`

The result is negative since the value is depreciating. We can therefore, conclude that the depreciation rate after 5 years of use is $1082.68.

b) What is the relative rate of change in the value of the machine after t years of use.

`\begin{gathered} \text{The relative rate of change in the value of machine after t years of use is} \\ \text{the first derivative of the function }V(t)\text{ which is} \\ V^(\prime)(t)=-8000e^(-0.04t) \end{gathered}`