Explanation:
10% per hour is tricky, because for the first hour it is 10% of the original 140 mg (14 mg). but for the second hour it is 10% of the updated level of 140 - 14 = 126 mg (12.6 mg). and so on.
so, we cannot simply say 5× 10% = 50%, therefore, it will take 5 hours.
no, instead we must use arithmetic sequences.
10% is represented as mathematical factor as 0.1.
so, when 10% are removed, it means that 90% remain.
90% = 0.9.
90% of 100% = 100% × 0.9
so,
a1 = 140
a2 = a1×0.9
a3 = a2×0.9 = a1×0.9²
an = a1×0.9^(n-1)
we need to find n for which an = 70 (half of the caffeine remains, meaning half of the caffeine is gone).
70 = 140×0.9^(n-1)
0.5 = 0.9^(n-1)
using the logarithm to the base of 0.9 to solve :
log0.9(0.5) = n - 1
n = log0.9(0.5) + 1
how to get log0.9(0.5) ?
all logarithms are related to each other :
loga(b) = logc(b)/logc(a)
in our case we can use ln (base e) or log (base 10) - whatever your calculator offers :
log0.9(0.5) = log(0.5)/log(0.9) = 6.578813479...
n = 6.578813479... + 1 = 7.578813479... hours
0.578813479...×60 = 34.72880874... minutes
so, after about 7.6 hours or 7 hours and 35 minutes half of the caffeine will be gone.