# calculus

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The population of a certain community is increasing at a rate directly proportional to the population at any time t. In the last yr, the population has doubled.
How long will it take for the population to triple?
Round the answer to the nearest hundredth, if necessary.

• calculus -

this fits the standard

A = a e^(kt), where a is the initial amount, A is the new amount, k is a constant and t is the time in suitable units.

in our case if a=1 , A = 2, t = 1
2 = 1 e^k
ln 2 = ln e^k
k = ln2

A = e^ (ln2 t)
when A = 3
3 = e^ ln2 t
ln2 t = ln3
t = ln3/ln2 = 1.58 days

or , since we have a case of "doubling"
N = 1 (2)^t

3 = 2^t
ln3 = ln 2^t
ln3 = t ln2
t = ln3/ln2 , as above

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