the population of a country is 65 millions if it grows exponentially in a rate of 1.5% annual
1.-calculate the estimate population in 12 years
2.-when the population increases 4 times
could somebody please explain me this kind of problems where you need to use the log and please tell me the answers
1. .015*65*12+65=76.7 million
2. 4*65=260 million
1. Multiply 65 million by (1.015)^12
(1.015^12 = 1.1956)
2. Multiply 65 million by 4.
You don't need to use log for the second one.
You can to 1. with a log table or with a hand calculator (as I did).
If you use logs:
Log (answer) = log (65 million) + 12 log 1.015
Then take the antilog
You will learn more if you do the calculations yourself than if I gave you the numerical answers.
Let
P = population in millions
t = time in years
Population growth is continuous. For continuous compounding use the formula:
P = 65*e^(.015t)
For the population in 12 years we have:
P = 65*e^(.015*12) = 65*e^(.18)
P = 77.8 million
To solve these types of exponential growth problems, we can use the formula for compound interest, which can be modified for population growth:
A = P(1 + r)^t
where:
A = final population
P = initial population
r = growth rate (as a decimal)
t = time period (in this case, measured in years)
1. To calculate the estimated population in 12 years:
Initial population (P) = 65 million
Growth rate (r) = 1.5% = 0.015 (as a decimal)
Time period (t) = 12 years
Using the formula:
A = 65,000,000(1 + 0.015)^12
To solve this, we can calculate the value inside the parentheses first:
(1 + 0.015) = 1.015
Then we raise that value to the power of 12:
(1.015)^12 = 1.197$ (rounded to 3 decimal places)
Now we can substitute this value back into the formula:
A = 65,000,000 * 1.197 = 77,805,000 (rounded to the nearest whole number)
Therefore, the estimated population in 12 years will be around 77,805,000.
2. To find when the population increases by 4 times:
We need to find the value of t when A/P = 4.
Using the formula, we can rewrite it as:
(1 + 0.015)^t = 4
To solve for t, we can take the logarithm (log) of both sides of the equation. In this case, we can use the natural logarithm (ln):
ln(1 + 0.015)^t = ln(4)
Then we can bring down the exponent using the logarithmic property:
t * ln(1 + 0.015) = ln(4)
To find t, divide both sides by ln(1 + 0.015):
t = ln(4) / ln(1 + 0.015)
Using a calculator, we find:
t ≈ 46.849
Therefore, the population will increase by 4 times in approximately 46.849 years.
Note: Logarithm is used to solve the equation in the second part of the question where we need to find when the population increases by 4 times. However, logarithm is not necessary for the first part where we only need to calculate the estimated population after 12 years.