In 1951, the population of India was 357 million people. By 1981 it had grown to 984 million. If the population is growing exponentially, when (in what month of what year) will the population reach 1 billion people?
let the equation be
P = 357(e)^(kt) where t is the time in years since 1951
so 1981 ---> t = 30
984 = 357(e^30k)
2.756303 = e^30k
30k = ln 2.756303
k = .0337963
so now we have P = 357(e^.0337963t)
so 1000 = 357(e^.0337963t)
2.80112 = e^.0337963t
t = 30.477
30 years + .477 of the next year
30 yrs + 5.7 months
or
in the 6th month of 1951
wouldn't it be past 1981 instead of in the 6th month of 1951?
of course, clearly a typo. Sorry about that.
since my t=30.477 and I defined as 1951 as t=0
the time is 1951 + 30.477 = 1981 + .477yrs.
Does the .477 years mean the 6th month (June) you were talking about?
To determine when the population of India will reach 1 billion people, we need to use exponential growth formula and solve for time. The formula for exponential growth is given by:
P(t) = P₀ * e^(rt)
Where:
P(t) is the population at time t
P₀ is the initial population
e is the base of natural logarithms (approximately equal to 2.71828)
r is the growth rate
t is the time passed
First, let's find the growth rate (r). To do this, we can use the population data you provided. We have the population in 1951 (P₀) as 357 million and the population in 1981 (P(t)) as 984 million. We can use these values to find the growth rate (r) using the formula:
r = (ln(P(t)) - ln(P₀)) / (t - t₀)
Where:
ln is the natural logarithm function
t₀ is the initial time
Using the given data, we have:
P(t) = 984 million
P₀ = 357 million
t = 1981 - 1951 = 30 years
Now, let's calculate the growth rate (r):
r = (ln(984) - ln(357)) / 30
Using a calculator, we find that the growth rate (r) is approximately 0.025574.
Next, we can use the exponential growth formula and solve for the time it takes for the population to reach 1 billion (P(t) = 1 billion). Let's rearrange the formula to solve for t:
t = (ln(P(t)) - ln(P₀)) / r
Plugging in the values:
P(t) = 1 billion
P₀ = 357 million
r = 0.025574
t = (ln(1000) - ln(357)) / 0.025574
Again, using a calculator, we find that t is approximately 62.09 years.
Since we started in 1951, we can add 62 years to get the final year:
1951 + 62 ≈ the year 2013.
Therefore, the population of India was projected to reach 1 billion people around 2013.