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?

How do you know it's the 6th month?

Thank you for your help!!

welcome!

To determine in which month the population will reach 1 billion people, we need to first find the growth rate. Exponential growth can be modeled by the equation P(t) = P0 * e^(rt), where P(t) represents the population at time t, P0 is the initial population, e is the mathematical constant approximately equal to 2.71828, and r is the growth rate.

Given that the population of India in 1951 was 357 million and in 1981 it was 984 million, we can calculate the growth rate by using the formula:

P(t) = P0 * e^(rt)

For the initial year (1951):

357 = 357 * e^(r * 0)
e^(r * 0) = 1 (any number raised to the power of 0 is equal to 1)
So, the equation simplifies to:
357 = 357

Next, we can use the second point (1981):

984 = 357 * e^(r * 30)
e^(r * 30) = 984/357
Simplifying further, we get:
e^(r * 30) ≈ 2.752
Taking the natural logarithm on both sides to solve for r:
r * 30 ≈ ln(2.752)
r ≈ ln(2.752)/30 ≈ 0.0549

Now that we have the growth rate (r), we can use it to find the time it takes to reach 1 billion people. Let P(t) be 1 billion (1000 million):

1000 = 357 * e^(0.0549 * t)
Dividing both sides of the equation by 357:
2.802 = e^(0.0549 * t)

To solve for t, we take the natural logarithm of both sides:
ln(2.802) = 0.0549 * t

Now to find the value of t, we divide both sides by 0.0549:
t ≈ ln(2.802)/0.0549 ≈ 37.34

So, the population of India will reach 1 billion people approximately 37.34 years after 1951. To find the specific month, we can calculate the number of months using the growth rate:

37.34 years * 12 months/year ≈ 448.08 months.

Hence, we add this to the starting month, January 1951:

January 1951 + 448 months ≈ May 1988.

Therefore, the population of India will reach 1 billion people around May 1988. It is not the 6th month; it is the 5th month (May). I apologize for any confusion in my previous response.