The population of the world was 7.1 billion in 2013, and the observed relative growth rate was 1.1% per year.
(a) Estimate how long it takes the population to double. (Round your answer to two decimal places.)
_______yr
(b) Estimate how long it takes the population to triple. (Round your answer to two decimal places.)
________yr
a=63.01
(a) find t where 1.011^t = 2
To estimate how long it takes for the population to double (a), we can use the formula for exponential growth:
P(t) = P₀ * (1 + r)ᵗ
Where:
P(t) is the population at time t
P₀ is the initial population
r is the relative growth rate
t is the time in years
In this case, P₀ = 7.1 billion, r = 1.1% (or 0.011 as a decimal), and t is what we want to find.
To double the population, we need P(t) to be 2 * P₀:
2 * P₀ = P₀ * (1 + r)ᵗ
Now we can solve for t:
2 = 1.011ᵗ
To solve for t, we'll take the log base 1.011 of both sides:
log₁.₀₁₁(2) = t
Using a calculator, we find that log₁.₀₁₁(2) is approximately 69.66. So, it would take approximately 69.66 years for the population to double.
Therefore, the estimated time for the population to double is approximately 69.66 years.
To estimate how long it takes for the population to triple (b), we use the same formula:
3 * P₀ = P₀ * (1 + r)ᵗ
Solving for t:
3 = 1.011ᵗ
Taking the log base 1.011 of both sides:
log₁.₀₁₁(3) = t
Using a calculator, we find that log₁.₀₁₁(3) is approximately 104.77. So, it would take approximately 104.77 years for the population to triple.
Therefore, the estimated time for the population to triple is approximately 104.77 years.