Acar was bought for 45000 in 2005. if the car loses 4.5% of its value semiannually  how much will it be worth at the end of 2017? in what year will the car be worth 0

To solve this we are going to use the compounded interest formula: 
where 
is the final value of the car after  years
is the price of the car
is the rate in decimal form
is the number of times the rate is compounded per year 
is the time in years

Part 1. Notice that since the car is losing 4.5% of its value semiannually, it is losing 9% annually; also the rate is going to be negative, so . Since the car its value semiannually, . Also, to find , we are going to subtract 2005 from 2017: . Finally, we know for our problem that . Now that we have all the vales we need, lest replace them in our formula:





We can conclude that at the end of 2017 the char will be worth $14,903.68

Part 2. Since we want to know the year in which the price of the car will be zero, . From our previous calculations we know that , ,and . Lets replace those values in our formula one more time:


Since  is the exponent, we are going to use logarithms to bring it down: 





Since  cannot be evaluated (you can't divide by zero in mathematics), we can conclude that the value of the car will never be zero.

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