I'm doing Principles of Physics, $10^{\text{th}}$ edition by Resnick, Halliday, Walker. I tried doing the following question.
At the end of a year, a motor car company announces that the sale of pickup trucks are down by $43.0 \%$ for the year. If sales continue to decrease by $43.0 \%$ in each succeeding year, how long will it tale for the sales to fall $10.0 \%$ of the original number?
My Attempt:
Let initial sales quantity be $a$. So according to the situation presented we are looking for the smallest $n$ that satisfies the following inequality: $$\begin{aligned}a\left(1-\dfrac{43}{100}\right)^n&\le\dfrac{10a}{100}\\ n\log_{10}\left(\dfrac{67}{100}\right)&\le\log_{10}\left(\dfrac{1}{10}\right)\\ n&\ge \dfrac{-1}{\log_{10}(67/100)}\approx 5.75 \text{ years}\end{aligned}$$
It would be great if someone could check my reasoning. I don't have the solutions manual or the answer key for this question. Thanks
Best Answer
You have the correct reasoning and method. You made a small mistake though. You wrote $$1-\frac{43}{100}=\frac{67}{100}$$when in fact, it is $\dfrac{57}{100}$. After you correct this, it should be fine.