Pretend the world's population in 1988 was 4.9 billion and that the projection for 2017, assuming exponential growth, is 7 billion.
What annual rate of growth is assumed in this prediction?
To determine the annual rate of growth, we can use the formula for exponential growth:
P(t) = P(0) * e^(rt)
Where:
P(t) is the population at time t
P(0) is the initial population
r is the annual growth rate
t is the time period
Given that the population in 1988 (P(0)) is 4.9 billion and the projected population in 2017 (P(t)) is 7 billion, we can plug in these values and solve for r:
7 billion = 4.9 billion * e^(r * 29)
Dividing both sides by 4.9 billion:
7 / 4.9 = e^(29r)
Simplifying further:
1.428571429 = e^(29r)
Taking the natural logarithm (ln) of both sides:
ln(1.428571429) = ln(e^(29r))
Using the property ln(e^x) = x:
ln(1.428571429) = 29r
Now, divide both sides by 29 to isolate r:
r = ln(1.428571429) / 29
Using a calculator, we can find:
r ≈ 0.0164 or 1.64%
Therefore, the annual rate of growth assumed in this prediction is approximately 1.64%.
To determine the annual rate of growth assumed in this prediction, we can use the exponential growth formula:
P(t) = P(0) * (1 + r)^t
Where:
- P(t) represents the population at time t
- P(0) represents the initial population (in 1988)
- r represents the annual growth rate
- t represents the number of years
Given that the population in 1988 (P(0)) was 4.9 billion and the projected population in 2017 (P(t)) is 7 billion, we need to find the value of the growth rate (r). We also know that the time span (t) is 2017 - 1988 = 29 years.
Substituting the values into the formula, we have:
7 billion = 4.9 billion * (1 + r)^29
To find the growth rate (r), we can rearrange the equation and solve for it. Here are the steps:
1. Divide both sides of the equation by 4.9 billion:
7/4.9 = (1 + r)^29
2. Take the 29th root of both sides:
(7/4.9)^(1/29) = 1 + r
3. Subtract 1 from both sides to isolate r:
(7/4.9)^(1/29) - 1 = r
To calculate this value, I will use a calculator:
r ≈ 0.0177
Therefore, the predicted annual rate of growth is approximately 0.0177, or 1.77%.
I assume you want to find the annual rate of growth. So,
4.9(1+r)^(2017-1988) = 7
r = 0.012375 or about 1.2%