Question Statement

The exponential growth rate of the population of a city is $1\%$ per year. After how many years will the population be doubled?

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Background and Explanation

Exponential growth occurs when the increase in a quantity is proportional to its current value. To find the doubling time, we use the growth formula $P(t)=P_0(1+r{)}^t$ and solve for the time $t$ required for the population to reach $2\times P_0$.

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Solution

To determine the doubling time, we follow these mathematical steps:

Step 1: Define the growth model

Given a growth rate of $1\%$ (or $0.01$), the population $P(t)$ after $t$ years can be modeled. If we assume an initial population of $100$: $P(t)=100(1.01{)}^t$

Step 2: Set the target population

We want to find the time $t$ when the population has doubled. If the starting population is $100$, the target population is $200$: $P(t)=200$

Step 3: Set up the equation

Substitute the target value into our growth model: $200=100(1.01{)}^t$

Step 4: Simplify the equation

Divide both sides by $100$ to isolate the exponential term: $2=(1.01{)}^t$

Step 5: Use logarithms to solve for $t$

To solve for the exponent $t$, we take the logarithm of both sides of the equation: $\log 2=t\log 1.01$

Now, isolate $t$ by dividing by $\log 1.01$: $t=\frac{\log 2}{\log 1.01}$

Step 6: Final Calculation

Calculating the values of the logarithms: $t\approx 69.66\text{ years}$

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Key Formulas or Methods Used

• Exponential Growth Formula: $P(t)=P_0(1+r{)}^t$
• Logarithm Power Rule: $\log (a^b)=b\log a$
• Doubling Condition: Setting the ratio of $\frac{P(t)}{P_0}=2$

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Summary of Steps

1. Set up the exponential growth equation using the $1\%$ rate ($1.01$ growth factor).
2. Set the equation equal to $2$ to represent the doubling of the initial population.
3. Apply logarithms to both sides to move the variable $t$ out of the exponent.
4. Divide $\log 2$ by $\log 1.01$ to find the number of years.