In 1987, the population of Mexico was estimated at 82 million people, with an annual growth rate of 2.5%. The 1987 population of the United States was estimated at 244 million with an annual growth rate of 0.7%. Assume that both populations are growing exponentially. When will Mexico double its population?
PMexico = 82(1.025)^t where t is the number of years past 1987
164 = 82(1.025)^t
2 = 1.025^t
take log of both sides
log 2 = log 1.025^t
log 2 = t(log 1.025)
t = log 2/log 1.025
= 28.07
so add 28 to 1987
To determine when Mexico will double its population, we need to use the exponential growth formula:
P(t) = P(0) * e^(rt)
Where:
- P(t) is the population at time t
- P(0) is the initial population
- e is the base of the natural logarithm (approximately 2.71828)
- r is the growth rate
- t is the time in years
We know that in 1987, Mexico's population was estimated to be 82 million, with a growth rate of 2.5% or 0.025. We need to find the time it takes for Mexico's population to double, so we will let P(t) = 2 * P(0).
2 * P(0) = P(0) * e^(0.025t)
Now, we can cancel out P(0) from both sides of the equation:
2 = e^(0.025t)
Next, we need to isolate t. To do this, we take the natural logarithm of both sides of the equation:
ln(2) = ln(e^(0.025t))
Since logarithm rules state that ln(e^x) = x, the equation simplifies to:
ln(2) = 0.025t
Now, divide both sides of the equation by 0.025:
t = ln(2) / 0.025
Using a calculator, we find:
t ≈ 27.73
Therefore, Mexico will double its population in approximately 27.73 years.
To find out when Mexico will double its population, we can use the formula for exponential growth:
P(t) = P0 * e^(r * t)
Where:
P(t) = population at time t
P0 = initial population
r = annual growth rate (as a decimal)
t = time in years
Let's apply the formula to Mexico's population:
P(t) = 82 million * e^(0.025 * t)
To find when Mexico will double its population, we substitute P(t) with 2 * P0 and solve for t:
2 * 82 million = 82 million * e^(0.025 * t)
Simplifying the equation:
2 = e^(0.025 * t)
To solve for t, we can take the natural logarithm of both sides:
ln(2) = ln(e^(0.025 * t))
Using the property that ln(e^x) = x:
ln(2) = 0.025 * t
Now, divide both sides by 0.025:
t = ln(2) / 0.025
Using a calculator, the approximate value of t is:
t ≈ 27.73 years
Therefore, Mexico is estimated to double its population after around 27.73 years from 1987.