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Sagot :
[tex]\begin{gathered} \text{Given} \\ V(t)=20000e^{-0.4t} \end{gathered}[/tex]
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.
[tex]\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}[/tex][tex]\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}[/tex]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.
[tex]\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}[/tex]
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