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If an initial quantity continuous decay at an exponential rate , then the final amount after a time is given by the following formula .
(a)
Find proportion of words remains unchanged.
The function is .
is the proportion of words that remain unchanged.
is the time since two languages diverged.
is the rate of replacement.
Since two languages diverged years ago .
Since rate of replacement is , .
Substitute and in .
.
Proportion of words remains unchanged is .
(b)
Find how many years will only 1% of the words remain unchanged.
Since 1% of the words remain unchanged, .
Substitute and in .
Apply logarithm on each side.
.
1% of the words remain unchanged in years.
(a) Proportion of words remains unchanged is .
(b) 1% of the words remain unchanged in years.
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