The value of a used car depreciates at a rate of 14 percent every year. If the car was bought for 15,000 dollars in February 2017, how much would it be worth in February 2023? Give your answer to the nearest 100 dollars.
Now, to help us solve this problem, what we’re gonna do is set up an exponential equation. And to do that, we have a general form. And that is that the function is equal to 𝐴, where 𝐴 is our initial amount, multiplied by 𝑏, where 𝑏 tells us about the rate that we’re looking at. But also we know that 𝑏 is always positive and not equal to one. And then this is raised to the power of 𝑥 where 𝑥 is the independent variable, which is usually time.
Well, in this problem, first of all, what we want to look at is what each of our parts are going to be. So, for instance, our 𝐴 is gonna be 15,000 because that’s the initial amount. And we know that the initial amount the car was bought for was 15,000 dollars. And then we know that our 𝑥 is gonna be equal to six. And the reason this is is because, in our problem, the independent variable is our time periods. And the time periods we’re looking at are years. And that’s because we’re told that the value of a used car depreciates at a rate of 14 percent every year.
Well, we know that the car was bought in 2017. And we want to find out its value in 2023. So therefore, our 𝑥-value is gonna be equal to six. And next, we have our 𝑏-value. And this tells us about the rate like we said. Well, we know that the rate in this problem is a depreciation of 14 percent. So therefore, our 𝑏-value is gonna be 0.86. But you might think, well, where did you get that from? Well, let’s think about how we get this.
Well, first of all, if we consider 100 percent, well, 100 percent means 100 out of 100 which is the same as one. Well, if we then subtract 14 percent from 100, and we do that because the car depreciates at a rate of 14 percent, well, then what we’re gonna be left with is 86 percent. Well, 86 percent means 86 out of 100. So we could write this as 86 over 100. Well, if we convert this into a decimal, so our decimal multiply, then what we’re gonna get is 0.86. So therefore, yep, our 𝑏-value is gonna be 0.86.
However, it’s also worth noting at this point something that would help you avoid a common mistake. And that is we got a decimal multiplier of 0.86. A common mistake is to get one of 1.14, and you get that by adding the 14 percent instead of taking it away. So always look out for the word here that we have which is “depreciates” because this means that we do subtract, not add. So now we can utilize the general form because our 𝑉, which is gonna be our value, so our function, is gonna be equal to 15,000 multiplied by 0.86 to the power of six, which is gonna give us 6,068.508 et cetera.
Well, it’s at this point we go, well, have we solved the problem? We found out the value of 𝑉, so the value of the car in February 2023. Well, no. Again, always check what the question wants you to leave your answer in. And here, it says, “Leave it to the nearest 100 dollars.” So therefore, what we can say is that the value of the car in February 2023 to the nearest 100 dollars is 6,100 dollars.