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Harrington is a large city with a population of about 350,000 . A recently taken census found that the annual growth rate of the city is 6%. If it continues to grow at this rate how many years will it take for the population to triple in size?

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After performing the calculation, we find that
\( t \) 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:


\[ P = P_0 * (1 + r)^t \]

Where:

-
\( P \) is the final population

-
\( P_0 \) is the initial population

-
\( r \) is the annual growth rate (as a decimal)

-
\( t \) is the time in years

We want the final population
\( P \) to be three times the initial population
\( P_0 \), so we can set
\( P = 3 * P_0 \). Plugging this into the formula gives us:


\[ 3 * P_0 = P_0 * (1 + r)^t \]

We can cancel
\( P_0 \) from both sides, since it is non-zero, and solve for
\( t \):


\[ 3 = (1 + r)^t \]

To solve for
\( t \), we take the natural logarithm of both sides:


\[ \ln(3) = \ln((1 + r)^t) \]

Using the power rule for logarithms, which says that
\( \ln(a^b) = b * \ln(a) \), we can simplify this to:


\[ \ln(3) = t * \ln(1 + r) \]

Now we can solve for
\( t \):


\[ t = (\ln(3))/(\ln(1 + r)) \]

We are given that
\( r = 6\% \) or
\( r = 0.06 \) as a decimal. Let's plug in the values:


\[ t = (\ln(3))/(\ln(1 + 0.06)) \]

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
(\( P_0 \)) = 350,000

- Annual Growth Rate
(\( r \)) = 6% = 0.06 (as a decimal)

- Final Population
(\( P \)) = Initial Population \(\times\) 3 (since we want it to triple)

Step 2: Set up the equation based on the exponential growth formula.


\[ 3 * P_0 = P_0 * (1 + r)^t \]

Step 3: Simplify the equation by dividing both sides by \( P_0 \), which gives us:


\[ 3 = (1 + r)^t \]

Step 4: Take the natural logarithm of both sides to solve for \( t \) (the time in years).


\[ \ln(3) = \ln((1 + r)^t) \]

Step 5: Apply the power rule of logarithms to bring \( t \) down as a coefficient.


\[ \ln(3) = t * \ln(1 + r) \]

Step 6: Isolate
\( t \) and solve for it.


\[ t = (\ln(3))/(\ln(1 + r)) \]

Step 7: Calculate the value using the annual growth rate as 0.06.


\[ t = (\ln(3))/(\ln(1 + 0.06)) \]

After performing the calculation, we find that
\( t \) 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.

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