After performing the calculation, we find that
is approximately 18.85 years. Therefore, it will take nearly 18.85 years for the population to triple in size if it continues to grow at an annual rate of 6%. This is a slightly more precise calculation compared to the previous estimation.
We can use the formula for exponential growth to find out how many years it will take for the population of a city to triple given a certain growth rate. The formula is:
![\[ P = P_0 * (1 + r)^t \]](https://img.qammunity.org/2024/formulas/business/high-school/1wyrnm3pkpj9n6f728eulitk4ygq0tmwoe.png)
Where:
-
is the final population
-
is the initial population
-
is the annual growth rate (as a decimal)
-
is the time in years
We want the final population
to be three times the initial population
, so we can set
. Plugging this into the formula gives us:
![\[ 3 * P_0 = P_0 * (1 + r)^t \]](https://img.qammunity.org/2024/formulas/business/high-school/mks31u5bzayxcz9r1ygxht48orc3w0d3ac.png)
We can cancel
from both sides, since it is non-zero, and solve for
:
![\[ 3 = (1 + r)^t \]](https://img.qammunity.org/2024/formulas/business/high-school/qpxn8tnm9h3ogomk4n6ebg5mb757b9yoer.png)
To solve for
, we take the natural logarithm of both sides:
![\[ \ln(3) = \ln((1 + r)^t) \]](https://img.qammunity.org/2024/formulas/business/high-school/o48qsmzq3fb5x15rcccftce0kz1lknzvkf.png)
Using the power rule for logarithms, which says that
, we can simplify this to:
![\[ \ln(3) = t * \ln(1 + r) \]](https://img.qammunity.org/2024/formulas/business/high-school/2ujottffwkj12dt22f8fj8b628spu1av9n.png)
Now we can solve for
:
![\[ t = (\ln(3))/(\ln(1 + r)) \]](https://img.qammunity.org/2024/formulas/business/high-school/p240orl2evg58jywcdkwk8zwi0atv10c5r.png)
We are given that
or
as a decimal. Let's plug in the values:
![\[ t = (\ln(3))/(\ln(1 + 0.06)) \]](https://img.qammunity.org/2024/formulas/business/high-school/nrk0ml5vos1tpjv74xtxz0r739e7zmqyit.png)
We'll calculate this step by step.
To calculate the number of years it will take for the population of Harrington to triple at an annual growth rate of 6%, we follow these steps:
Step 1: Identify the given values.
- Initial Population
= 350,000
- Annual Growth Rate
= 6% = 0.06 (as a decimal)
- Final Population
= Initial Population \(\times\) 3 (since we want it to triple)
Step 2: Set up the equation based on the exponential growth formula.
![\[ 3 * P_0 = P_0 * (1 + r)^t \]](https://img.qammunity.org/2024/formulas/business/high-school/mks31u5bzayxcz9r1ygxht48orc3w0d3ac.png)
Step 3: Simplify the equation by dividing both sides by \( P_0 \), which gives us:
![\[ 3 = (1 + r)^t \]](https://img.qammunity.org/2024/formulas/business/high-school/qpxn8tnm9h3ogomk4n6ebg5mb757b9yoer.png)
Step 4: Take the natural logarithm of both sides to solve for \( t \) (the time in years).
![\[ \ln(3) = \ln((1 + r)^t) \]](https://img.qammunity.org/2024/formulas/business/high-school/o48qsmzq3fb5x15rcccftce0kz1lknzvkf.png)
Step 5: Apply the power rule of logarithms to bring \( t \) down as a coefficient.
![\[ \ln(3) = t * \ln(1 + r) \]](https://img.qammunity.org/2024/formulas/business/high-school/2ujottffwkj12dt22f8fj8b628spu1av9n.png)
Step 6: Isolate
and solve for it.
![\[ t = (\ln(3))/(\ln(1 + r)) \]](https://img.qammunity.org/2024/formulas/business/high-school/p240orl2evg58jywcdkwk8zwi0atv10c5r.png)
Step 7: Calculate the value using the annual growth rate as 0.06.
![\[ t = (\ln(3))/(\ln(1 + 0.06)) \]](https://img.qammunity.org/2024/formulas/business/high-school/nrk0ml5vos1tpjv74xtxz0r739e7zmqyit.png)
After performing the calculation, we find that
is approximately 18.85 years. Therefore, it will take nearly 18.85 years for the population to triple in size if it continues to grow at an annual rate of 6%. This is a slightly more precise calculation compared to the previous estimation.