A population grows exponentially at a rate of 2.6% per year. How long will it take for the population to triple?
when 1.026^x = 3
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To find out how long it will take for the population to triple, we need to use the exponential growth formula:
P(t) = P₀ * (1 + r)^t
Where:
P(t) is the population at time t
P₀ is the initial population
r is the growth rate
t is the time
In this case, we know that the growth rate is 2.6%, which can be written as 0.026 as a decimal, and we want to find the time it takes for the population to triple, so P(t) = 3 * P₀.
Substituting these values into the formula, we have:
3 * P₀ = P₀ * (1 + 0.026)^t
Dividing both sides of the equation by P₀, we get:
3 = (1 + 0.026)^t
Taking the natural logarithm (ln) of both sides, we obtain:
ln(3) = ln((1 + 0.026)^t)
Using the logarithmic property of ln(a^b) = b * ln(a), we can rewrite the equation as:
ln(3) = t * ln(1 + 0.026)
Now we can solve for t by dividing both sides of the equation by ln(1 + 0.026):
t = ln(3) / ln(1 + 0.026)
Using a calculator, we can find that ln(3) ≈ 1.0986 and ln(1 + 0.026) ≈ 0.0253. So:
t ≈ 1.0986 / 0.0253
Calculating this expression, we find that t ≈ 43.44 years.
Therefore, it will take approximately 43.44 years for the population to triple at a growth rate of 2.6% per year.