If the population of the country is growing 3.7% compounded contionously.
Quaterly. how long will it take the population to triple
Use continuous compound formula given two problems ago,
A=PeRT
Here
R=0.037
P=1
A=3
find T.
To find out how long it will take for the population to triple, we can use the continuous compounding formula. The formula for continuous compounding is given by the equation:
A = P * e^(rt)
Where:
A = Final amount (in this case, tripling the population)
P = Initial amount (current population)
r = Annual growth rate in decimal form
t = Time in years
e = Euler's number (approximately 2.71828)
In this case, we need to solve for t. Since the population growth is given quarterly, we'll need to convert the annual growth rate of 3.7% to the quarterly rate.
To convert an annual rate to a quarterly rate, we divide the annual rate by 4. In this case, the quarterly growth rate (r) would be 0.037/4 = 0.00925.
Now, let's plug in the numbers to solve for t:
3P = P * e^(0.00925t)
Next, we simplify the equation by dividing both sides by P:
3 = e^(0.00925t)
To solve for t, we take the natural logarithm of both sides:
ln(3) = ln(e^(0.00925t))
By using the property of logarithms, we can bring down the exponent:
ln(3) = 0.00925t * ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(3) = 0.00925t
Finally, we divide both sides by 0.00925 to solve for t:
t = ln(3) / 0.00925
Using a calculator, we find:
t ≈ 75.15
Therefore, it will take approximately 75.15 years for the population to triple.