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.

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

To find out how long it will take for the population to triple, we need to solve a differential equation that represents the rate of growth of the population.

Let P(t) represent the population at time t. Since the population is increasing at a rate directly proportional to the population at any time t, we can express this relationship using the equation:

dP(t)/dt = k * P(t),

where k is the proportionality constant.

To solve this differential equation, we can separate variables and integrate both sides:

dP(t)/P(t) = k * dt.

Integrating both sides gives:

∫(1/P(t)) dP(t) = ∫k dt,

which simplifies to:

ln|P(t)| = kt + C,

where C is the constant of integration.

To find C, we can make use of the given information that the population has doubled in the last year. Let's assume the population at time t=0 is P₀:

ln|P₀| = C.

Substituting this back into the equation, we have:

ln|P(t)| = kt + ln|P₀|.

Now, let's consider the original question. We want to find out when the population will triple. This means we are looking for a time t for which the population will be 3 times the initial population, P₀.

So we have the equation:

ln|3P₀| = kt + ln|P₀|.

Now, we can solve for t:

kt = ln|3P₀| - ln|P₀|,

kt = ln|(3P₀)/P₀|,

kt = ln|3|,

t = ln|3| / k.

Since we don't know the specific value of k, we can't determine t exactly. However, we can still provide an approximate answer by using the given information that the population has doubled in the last year.

Given that the population has doubled in the last year, we can write:

2P₀ = e^k,

which implies:

k = ln(2).

Now, substituting this value of k:

t = ln|3| / ln(2).

Using a calculator, we find:

t ≈ 1.585.

Therefore, it will take approximately 1.59 years (rounded to the nearest hundredth) for the population to triple.