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A car that when purchased 5 years ago cost $5,000 has a value now of $900. Find the value of the car of 8 years after its purchase by using the exponential model V(t) = V0etb, in which V(t) is the value of the car at any time t, V0 is the initial cost, t is the time in years, and b is the rate of depreciation. Round your answer to the nearest hundredth.

Answer :

We will first calculate the value of b using the given infomation:

[tex]V(5)=V_0\cdot e^{5\cdot b}[/tex]

[tex]900=5000\cdot e^{5\cdot b}[/tex]

[tex]\frac{900}{5000}=e^{5\cdot b}[/tex]

[tex]\frac{9}{50}=e^{5\cdot b}[/tex]

[tex]b=\frac{1}{5}\log (\frac{9}{50})[/tex]

That must be the rate of depreciation. Now lets find the value of the car 8 years after:

[tex]V(8)=5000\cdot e^{\frac{8}{5}\cdot\log (\frac{50}{9})}[/tex]

[tex]V(8)\approx321.67[/tex]

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