A ball is dropped from a height of 1.60 m and rebounds to a height of 1.40 m. How many rebounds will the ball make before losing at least 90% of its energy?

The energy of the ball is proportional to the height it reaches (PE = mgh). So since after one bounce the ball reaches 87.5% of its original height (1.4m/1.6), and assuming that the ball losses a fixed percentage of its remaining energy with each bounce, you can calculate how many bounces it takes to have only 10% of its original energy left by solving for n:

(1.4/1.6)^n <= 0.1

Do you know how to solve this for n? You can either use logarithms or just guess at values of n and plug them into your calculator until you find one that works.