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The value of a used car can be modeled by the formula V=Vo(1-r)^t where Vo is the car's purchase price, in dollars; r is the car's constant annual rate of decrease in value, expressed as a decimal; and V is the car's dollar value at the end of t years. A used car has a constant annual rate of decrease in value of 0.075. According to the model, what expression would give the number of years after purchase for the car to reach a value that is 50% of its purchase price?

Answer :

Following the equation

[tex]V(t) = V_0(1-r)^t[/tex]

We start with an initial price of

[tex]V(0)=V_0[/tex]

and we're looking for a number of years t such that

[tex]V(t)=\dfrac{V_0}{2}[/tex]

If we substitute V(t) with its equation, recalling that

[tex]r = 0.075 \implies 1-r = 0.925[/tex]

we have

[tex]V_0\cdot (0.925)^t=\dfrac{V_0}{2} \iff 0.925^t = \dfrac{1}{2} \iff t = \log_{0.925}\left(\dfrac{1}{2}\right)\approx 8.89[/tex]

So, you have to wait about 9 years.

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