# Finding Time for Sales to drop below $10 \%$.

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

• How is this question from a physics book? – YuiTo Cheng Apr 28 at 13:38
• It's from the chapter Measurement which is more or less about Mathematical Modelling of presented situations. :) – Paras Khosla Apr 28 at 13:44

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.