+1207 A ball bounces back up \(\frac{2}{3}\) of the height from which it falls. If the ball is dropped from a height of 243 cm, after how many bounces does the ball first rise less than 30 cm?
+129852 We need to solve this
243 (2/3)^x = 30 divide both sides by 243
(2/3)^x = 30/243
(2/3)^x = 10/81 tak the log of both sides
log(2/3)^x = log (10/81) and we can write
x log(2/3) = log (10/81) divide both sides by log(2/3)
x = log (10/81) / log(2/3) ≈ 5.16 = after the 6th bounce
Check:
After the 5th bounce the ball will rise to 243(3/4)^5 = 32 cm
After the 6th bounce the ball will rise to 243(2/3)^6 = 21.3 cm
