Question

A certain type of fluorescent dye used in scientific research loses its brightness over time when exposed to light. Initially, the brightness is at 100% of its original intensity. Each hour, the brightness decreases to 80% of the intensity of the previous hour. The recursive formula describing this decrease is
B
(
n
)
=
B
(
n
−
1
)
⋅
0.80
, where
B
(
n
)
is the brightness as a percentage of the original intensity after 𝑛 hours, and
B
(
0
)
=
100
. Determine the number of hours it will take for the dye to reach 10% of its original brightness.

Answer 1

We have B(n) = 100(0.8)^n. Set B(n) = 10 and solve:

100(0.8)^n = 10
(0.8)^n = 0.1
n = ln(0.1) / ln(0.8) ≈ 10.32 hours.

So it takes about 10.32 hours to reach 10% exactly. If you count whole hours, after 10 hours it's ≈10.74% and after 11 hours it's ≈8.59%, so you need 11 full hours to drop to 10% or below.