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# interest

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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