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Ben bought a car for $35,000 in 2011. The car depreciates at a constant rate of 18% per year. How long will it be until the depreciated cost of the car is worth $12,000?

 Jan 14, 2020
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There's a couple of ways to do this problem, I'll do a rough-estimate solution and an algebraic one.

 

A simple solution to estimate how long it will be until the car's value is $12,000 is to simply depreciate the car's current value year-by-year by 18%, or 82% of its current value until the car is worth less than $12,000. In other words, we can keep multiplying the car's value by 0.82 until it is less than $12,000:

2011: $35000, 2012: $28700, 2013: $23534, 2014: $19297.9, 2015: $15824.3, 2016: $12975.9, 2017: $10640.2

Since its value in 2017 is now less than $12000, we know that it would take between 5-6 years for the depreciated car's value to be worth $12,000.

 

An exact solution will require an inkling of algebra. We can model what the car will cost, \(C\), at any point in \(t\) years given the initial price, \(P\), and decay rate, \(R\), using an exponential formula \(C = P * (1-R)^t\). Plugging in our givens, we have \(C = 35000(0.82)^t\). We want to know when the depreciated cost of the car is worth $12000, so we'll be solving for \(t\) with a fixed \(C\). Mathematically, we have \(12000 = 35000(0.82)^t\), which can be solved as follows:

\(12000 = 35000(0.82)^t\)

\(\frac{12}{35}=0.82^t\)

\(\log_{0.82} (\frac{12}{35})=t\)

\(t \approx 5.39399\)

So, we can see that it would be about 5.4 years from 2011 until the car's value is worth $12,000. This answer agrees with our rough estimate of between 5-6 years in our earlier approximation. Hope this helps :)

 Jan 14, 2020

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