What formular do i use for this question?

Question

An exponentially growing population triples in size in 3 years. How long does it take the double in size?

Answer 1

(1) Let's say the population at the start is P0

(2) Since the population rise is exponential, the population function, as a function of years (y), can be defined as

P(y) = P0*b^y, where the base b is yet to be determined

(3) Since you've stated that P(3) = 3*P0, we can determine b as follows:

P(3) = 3*P0 = P0*b^3, so we see that b^3 = 3

(4) Taking the natural log (ln) of the last expression, we have

ln(b^3) = 3*ln(b) = ln(3), then . . .

(5) ln(b) = ln(3)/3 = 0.366204, so b = e^0.366204 = 1.44225

(6) Therefore. P(y) = P0*(1.44225)^y

(7) To find the value of y when (1.44225)^y = 2, take the ln of both sides and solve for y, i.e.:

y*ln(b) = ln(2), so y = ln(2)/ln(b) = 0.693147/0.366204 = 1.893 years

Answer 2

You're gonna want to have e in there. So:

Since it triples in 3 years we'll write:

3p=pe^(3r) Where p is the starting amount and r is the growth rate

Then dividing by p we get:

3=e^(3r)

Take the ln of both sides:

ln3= 3rlne

so

ln3=3r

r=(ln3)/3

To double we have:

2p=pe^((ln3)/3*t) (by plugging (ln3)/3 into pe^(rt))

divide by p

2=e^((ln3)/3*t)

take the ln

ln2=((ln3)/3*t)lne

ln2=((ln3)/3*t)

multiply by 3

3ln2=t*ln3

so

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

Answer 3

The exponential function is:

f(t) = 3^(t/3)

In year 0 (t=0):

f(0) = 3^(0/3) = 3^0 = 1 (original population multiplier)

In year 3 (t=3):

f(3) = 3^(3/3) = 3^3 = 3 (triple the population)

You want to find the value of t where the population is 2 (double):

f(t) = 3^(t/3) = 2

Take the log of both sides:

log(3^(t/3)) = log(2)

Remember this rule of logs --> log(x^a) = a log(x):

(t/3) log(3) = log(2)

Divide by log(3)

t/3 = log(2) / log(3)

Multiply by 3:

t = 3 log(2) / log(3)

Get out your calculator...

t = 1.89278926

Answer:

Approximately 1.9 years (which is about 1 year 11 months)

Answer 4

Ae^t/3=2A

t/3 =ln(2)

t=2.079441542 years

Steed: Its per unit time, if a population started out at time 0 as 100 and another as 300 as time 3 then the function is 100(3)^1/3x